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Space is not perfectly empty. There are some sparse atoms and particles, which, under the right conditions, can be used to exert a force and use Newton 3rd law.

If you want to use fluid dynamic forces in space, you just need big wings. How big?

This answer gives an estimate:

Because of the low density of particles in the interstellar medium, a normal-sized wing will not act as an airfoil. It will be more like bouncing tennis balls off the wing every so often.

The Knudsen number quantifies when fluid dynamics takes over: Kn = mean_free_path / airfoil_length_scale. Knudsen numbers greater than about 10 are in the regime of ballistic collisions, rather than fluid flow. The mean free path in the interstellar medium is huge (about 70 astronomical units). So to get fluid dynamics, you'd need an airfoil of size more than the Sun–Jupiter distance.

Negative mass.

Each of your creatures contains a precious nugget of negative mass. When one of these creatures reproduces, the offspring are each given a fragment of the nugget. As they travel the universe, each creature accumulates any additional negative mass encountered and adds it to the nugget. When a creature dies, the nugget is claimed by its kin.

Negative mass can be used to propel. The larger and more unmassive this chunk is, the more useful it is for propulsion.

https://en.wikipedia.org/wiki/Negative_mass

For two positive masses, nothing changes and there is a gravitational
pull on each other causing an attraction. Two negative masses would
repel because of their negative inertial masses. For different signs
however, there is a push that repels the positive mass from the
negative mass, and a pull that attracts the negative mass towards the
positive one at the same time.

Hence Bondi pointed out that two objects of equal and opposite mass
would produce a constant acceleration of the system towards the
positive-mass object,[4] an effect called "runaway motion"...

Such a couple of objects would accelerate without limit (except
relativistic one); however, the total mass, momentum and energy of the
system would remain 0.

The nugget would be kept in the center of the body. When needing propulsion, the nugget would be moved to the end of an appendage. There it would repel the main body of the creature and be attracted at the same time. The same principle could be used to brake or steer by moving the nugget to a different position.

The creature is actually a giant bussard ramjet. It just looks like it's swimming, because the flaps are used to move particles from the surrounding area towards the body, where they are collected by polyp-like areas covering the body. They are then moved towards an organ that holds the particles until they are required for energy generation or used as propulsion matter. The body could also emit an EM field that moves particles closer to the body, so that the flaps can collect more of it.

The end result would be a movement similar to the breaststroke of a swimmer with a forward movement of the flaps close to the body to avoid pushing particles away and a wide backwards movement to collect a large volume of particles.

It's probably not scientifically viable, because the movement would most likely expand more energy than it could generate from what it collects, but it's at least science based. You could make the flaps solar energy collectors and have it hibernate during times where it's too far away from a sun to collect energy, if you want it's energy consumption to be slightly more realistic.

Surprisingly a creature swimming in space would be possible, in the sense that it would be able to change its position/location, but not is momentum which would make it useless for long distance travel.

The concept is called swimming in spacetime and is the only reactionless propulsion idea we have that would not violate physics.

According to wikipedia:

“Swimming in spacetime” is a geometrical motive principle that exploits the curved spacetime metric of the gravitational field to permit an extended body undergoing specific deformations in shape, to change position. In weak gravitational fields, like that of Earth, the change in position per deformation cycle would be far too small to detect, but the concept remains of interest as the only unambiguous example of reactionless motion in mainstream physics.
https://en.wikipedia.org/wiki/Reactionless_drive#Movement_without_thrust

By my understanding, it works through the fact the gravitational field is a gradient (slightly different in different positions, e.g. weaker further away from the source.). By shaping the body so that the center of mass of that body is in a different location where the gravitational field is stronger or weaker and then going back to the original shape, the body would have changed location ever so slightly compared to the previous location. This, however, does not impart any momentum on the body, as the force necessary to take on the shape is exactly the same force as the force to go back to the original shape (That's how I understood the theory, but the more I think about it, the forces required to change shapes should be different in different gravities.).

Here's a question on stackexchange on the topic: https://physics.stackexchange.com/questions/886/swimming-in-spacetime-apparent-conserved-quantity-violation
Here is a paper on the topic: http://science.sciencemag.org/content/299/5614/1865

Perhaps gravity, like that of the space probes. Or they could wear suits of magnets that attract or repel a planets or asteroids magnetic field. If they are some kind of animal perhaps they have a developed gland for an electrical field like an eel. Another idea is that they detect and capture neutrinos from the photons from stars and use them as a connective tissue or string. There are probably many different ideas of how they move because I’m pretty sure they cannot sit still in space. I would probably go with some type of jet stream orbit, and here is where their dust food would accumulate anyway.

They are not actually swimming - their extremities are actually EM drives which allow for reactionless motion.

Sorry but you will never be able to swim in there. One m³ of "space" will weight in order 10^-21 to 10-25 kg. Your ordinary air is 1.225kg/m³. There is like 10¹⁹ grains of sand on Earth. You will destroy your body or run out of food/energy way before you produce some decent speed. Maybe use something like an Ion engine.

I can think of two methods of moving through space:

1. 
   Gravity assist—the creatures are highly attuned to gravity wells and know how to position themselves perfectly to slingshot around the universe.


2. 
   Pooping near the speed of light. If you just shoot a tiny poop at 0.999999C, then the energy is ridiculous enough for the "equal and opposite reaction" of Newton's laws to push your creature forward very well indeed.

Actually moving fins against particles in space wont do enough, unless your creature weighs ~1 gram (but that's kinda silly, we'd be moving into fiction from here)

Ok one more bonus point:

3. 
   Super-giant-colossal squid: This is similar to gravity assist, but with more control—tentacles are galaxy-sized in length, and at the end of them is where most of the creature's weight is. By moving the tentacles around, they can shift their weight around closer and farther from various sources of gravity.

They are not actually swimming.

They are moving extremities in order to manipulate angular momentum. This allows them to change orientation.

See for example https://en.wikipedia.org/wiki/Falling_cat_problem

Creature can emit light (or other form of radiation) from its skin cells, e.g. like light bugs or glowing sea creatures.

Emitting light pushes the creature in opposite direction.

Each cell can only emit light for a brief period of time. It takes time to prelenish the chemicals involved, and keeping the light on for too long will cook the cell.

So as the creature sends a wave through its body, it activates lights on the part of the wave that is facing backward. It could probably achieve same effect by spinning, but either it gets dizzy, or it wants to keep its head pointed forward.

Strictly speaking, with the size you have provided and the requirement of "Swimming" (Propulsion through self movement), what you are asking for is impossible. Swimming on our planet only works because we have things of sufficient density to displace to move ourselves forwards. Solar sails don't fit your criteria because they aren't active transportation. There are no active methods that you can use in the vacuum of space to propel yourself with no give/take.

What I would recommend is making this amoeba/whale like a balloon. A thin amoeba shell over its preferred source of food and using the byproducts of digestion to propel slightly. It might take 100000000000 years to reach the next food source at sufficiently low speeds but you would eventually make it to something. maybe.

The things that would work against you at this stage are :

A. Gravity. It would take quite a bit of speed to get out of the solar systems and the eventual pull of the suns.

B. Debris. It may be far and between but there is a good chance of eventually being punctured by debris given the distance and time.

So what discoveries in physics since Newton will enable them to move this way?

NONE

But don't worry, I got you covered.

Your space whale has a huge mouth and baleens that can collect the interstellar particules. Theses are then stocked as food and propellant.

Your whale has a special organ that can expel any matter at very high speeds (read a ion engine). This organ needs electricity to function.

That's where the huge flippers come into play. The space whale start moving its gigantic flippers and using a dedicated organ, stop them almost instantly when they reach top speed.

The moment of inertia energy is converted into electricity used by the organ that actually propel your whale.

Hence, the whale move by pulse and appear to swim into space, leaving a trail of space dust behind it. It goes into hibernation for long periods of time, in order to get enough propellant to accumulate in its mouth.

They are swimming in space. It's just that their local space is not as much of a near vacuum as our solar system or most of the interstellar medium. The local space happens to be the denser sections of a large, heavy accretion disk that has yet to fuse into a star or form planets. Or they swim when they pass through a binary star system, one star eating the other, with matter streaming from one point to the other giving them enough high speed, low density material to move against.

If you're attached to them being interstellar, perhaps they return to the "stream" to mate, and the dancing is a ritual toward that goal, forming eddies and whorls that they can sense and find each other. Add "seeking love" and you satisfy your thematic requirements that they dance, and that the act can been seen as beautiful.

The 'swimming' motions are actually just used to pump whatever reaction-mass through the creature's body, expelling out the 'rear'.

It is obvious that they will not be the size of a solar system.

This means they cannot have fins large enough to be useful in the vacuum of space and cannot swim that way.

But if they had a naturally occurring powerful em field projected from their flippers perhaps they could use that em field to act as an invisible fin. Using it to push the atoms in space Around to propel this leviathan through space.

How they are able to generate an em field large enough to effect a sufficient mass to use as propellant through space I leave to you.

But it would work.

How about ion thrusters?

The creatures inhale and store relatively small amount of xenon and due to internal biology(the wings are solar collectors?) produce enough electricity to to shoot xenon ions through some orifice in their body to produce thrust. I don't have the number on hand, but I understand that xenon ion thrusters are extremely efficient but they have very low acceleration

While this isn't "swimming", the combination of wings (for solar) and thrust might look enough like swimming/flying to the casual observer

If you want to get more speculative, they swim through the "Luminiferous aether".

The aether was a theoretical medium through which light was thought to propagate, thus explaining its wave-like nature. The theory was discarded as ever-more-sensitive experiments failed to detect any ripples in the apparent speed of light caused by the momentum of Earth not matching the momentum of the aether...

Of course, we have since discovered that all of the elementary particles of matter also travel as waves. (This is how electron microscopes work) The wavelength merely decreases as the particles get larger until it is too short for us to detect. Which would offer a theoretical alternative explanation for why the experiments detected no variation between light and matter due to aetheric ripples: The test equipment itself is also an aetheric ripple and therefore there can never be a difference in medium velocity between light and matter.

Your creatures have found some way to push against the aether itself. (Most likely the creatures are actually the ripples caused by the actions of a higher-level entity that is not entirely perceptible to our senses.) This would allow them to "swim" through seemingly empty (ripple-free) space.

There are a few potential side-effects to this:

Such creatures could be a source of extra-universal energy to keep the universe running (similar to the theorized, but never discovered "white hole").

Such creatures would be theoretically capable of exceeding the speed of light.

Such creatures would likely be harmed only by (from our point of view) exceedingly violent events. (If all matter is ripples in the aether, they are the water-bug swimming on the surface. It would take some really big ripples to drown them.)

Such creatures might appear spontaneously in relatively calm places and disappear in turbulent places. (The swimmers get out of the pool when the water gets too churned up.)