Swimming in space So we know you can't "swim" (or fly) in space as creatures can do on Earth, with no medium to push against, hence you need to expel something to get some propulsion. 
But my question is about whether you could perform a sort of clumsy swimming type movement, without expulsion, using the following method. 
Let's say you're floating in space, and you kick out your leg. Wouldn't this put you into a sort of spinning motion (because of Newton's 3rd law of motion)? Now, imagine, you swing your arms, this would put you in a different spinning movement, perhaps around a different axis. 
But wouldn't the combination of two different spinning motions, each with a different axis, cause you to start moving?
Now the probable answer to this is, the axis of the spin is always going to be the centre of gravity. But if you could somehow change your centre of gravity between each action by either changing shape or you somehow had the ability to move mass around within your body (and I'm aware this would cause an additional invocation of Newton's 3rd law), would this help?
 A: Mind if I pick a few nits before telling you why this won't work to make you move?
Not hearing any objections, so I'll nitpick away. Note that the center of gravity (center of mass is also a better term) is a point, the axis of spin passes through it but the axis is a line, so avoid saying "the axis is this point"; it just sounds weird.
But all of that is moot because your clumsy swimming method is fundamentally flawed. You forgot one important invocation of Newton's 3rd Law. Let's run through the process.
You kick out your leg. As you correctly pointed out, the 3rd law implies that the force you use to kick out your leg also causes the rest of you to begin rotating. If you do nothing else you should keep rotating, right? Yes! If you do nothing else, you should keep rotating. But now comes the experimental part. Try kicking out your leg without doing anything else for like a minute. Bet you can't do it. Why? Because at some point, your leg can't go any farther and stops. Importantly, the act of stopping your leg means your body imparted a force on it. Just as the original force to move your leg began a rotation of your body, the force required to stop your leg cancels the rotation of your body.
The important thing to consider here is that both momentum and angular momentum must remain conserved. If you are sitting isolated in space; not moving or rotating and you don't expel or interact with any material, then your momenta are conserved. You can move or rotate your leg or arm in one direction to make the rest of your body move or rotate in the opposite direction, but once you try to bring your arm or leg back to a rest position, that cancels out the movement or rotation, leaving you with zero momentum again. What you'll find is that you'll even stay oriented in the same direction when you bring your leg back to rest position. This is because there can't be any average angular momentum over a period of time if there was none to begin with. It'd be the most frustrating clumsy swim of your life.
On Earth, there's plenty of sources of friction and other types of resistance to trick you into thinking this should work in space, but it doesn't. And if I ever become friends with an astronaut, I'll ask them to make a video during an EVA that shows how futile swimming in space is (That's gotta be a huge favour to ask of them. Space is like an enormous bottomless pit on all sides; one slip and you're gone. Can you say "agoraphobic nightmare"?)
A: Yes, we can.
All these answers stating that it's not possible are restricting themselves to Newtonian mechanics.
It turns out that, once you take General Relativity effects (or at least space curvature) into account, "swimming in curved spacetime is indeed possible", according to a result published last year (2016) [1], which built on ideas from a paper in Science, from 2003 [2].
[1] https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.121502  ;
https://arxiv.org/abs/1611.06183
[2] http://science.sciencemag.org/content/299/5614/1865
A: You should question yourself on what is Swimming? The answer is contained in your question. You mentioned "... you're floating in space, and you kick out your leg", so clearly as per your question, your kicking "something". In earth, in water your kicking the water, which in turns pushes you forward. 
But what about space? What can you kick? It's absolute vacuum. If you're entirely isolated from all mass in the universe, there would be no more than 10 molecules per cubic centimetre and in earth, you will find 1.5 sextillion molecules in a drop of water (0.05mL).
So technically you would move (even if there was no initial propulsion) but not as fast as you're on earth. It's hard to imagine how it would be, as you have never experienced being in vacuum. But your withers and swings would, yes, make you move, but probably not in the direction you wish and definitely not as fast as you would move in water.
And consider buying a space suit with additional oxygen capacity of more than 12 hours before you start experimenting this 'Space Swim'. 
A: I agree with the other comments on here that you would not be able to achieve anything close to "swimming" with your body movements. But considering your body's attitude in space, it may respond similar to a reaction wheel. A reaction wheel is something that some satellites use to adjust their attitude without expending fuel.
https://en.wikipedia.org/wiki/Reaction_wheel
