This is weird: if the OP is on the east coast of the US of A, I was having The Exact Same Thought at almost the exact same time.
I was in a coffee shop, luckily equipped with plenty of things bolted down that I could grab onto. My first thought was to propel myself toward the front door to block it, so no one would wander outside and float away.
I’m recalling a fascinating article I read on the topic. The key problem is the air’s not viscous enough at our mass & speed scale AKA the Reynolds number is too low.
But I can’t remember where I found it. If I do I’ll post it.
In the meantime, here’s a bit on swimming in water under reduced but still non-zero G: Lunar Swimming.
I think the bigger problem would be the air’s lack of density, not its lack of viscosity. In liquids, at least, swimming speed is largely independent of viscosity (though I think the swimmers get tired quicker in high viscosity).
Given a large enough zero-g enclosure, though, you could eventually overcome that and build up speed.
Mythbusters tested this and concluded that swimmers swim about the same speed in more viscous liquids. But I have serious doubts about whether the result applies equally to “swimming” in a gas at zero gravity, which is a very different mode of locomotion.
Much better than strapping stuff to your arms is to make some big-ass swim fins and kick with your legs. Humans legs are much stronger than arms.
One big problem is that the world’s oceans are no longer kept in place. What are you going to do when you’re floating around and a blob of water the size of Wisconsin drifts towards you? Expect lots of drownings from runaway water balls.
One of the issues is that conventional swimming takes place at the air/water interface, where you have two mediums of vastly different viscosity & density.
Swimming strokes that let the arms recover in the air, e.g. crawl, backstroke, butterfly, have a massive inbuilt advantage over strokes that require the arms to recover underwater, e.g. breast stroke.
Swimming in zero G air is more like the breaststroke case than the other cases. It’s darn hard to make enough thrust on the thrust stroke to avoid offsetting substantially all of it on the recovery stroke. Only the net of the two contributes to useable acceleration. Hence the value of things like the flutter kick with mongo fins; both strokes are thrust strokes, albeit of different efficiencies.
Well, many airplanes can actually fly upside down, so I’m gonna guess that they’d work okay in 0G. For those kinds of planes that can’t fly upside down, I don’t have any idea!