Whataboutism for is for sure one of the most common these days. It's also what I'd call a red herring. I find it interesting that this fallacy has traditionally been used by the Russians going all the way back to the Soviet era and continuing with Putin. Every time they are criticized for anything, it's "what about racism in the United States."
I would also mention something that I think plagues political discussion everywhere: the tendency to use anecdotal evidence in an attempt to prove a wider point. The closest formal logical fallacy is the fallacy of composition, concluding that what is true of the part is true of the whole. This means over-generalizing from the behavior of individuals to that of the group, whether it is Muslims, police, liberals, conservatives, blacks, whites, you name it.
Then there's classic straw mannery. Which aggravates me so much that I tend to disengage when I see people willfully or ignorantly mischaracterizing my arguments.
As to which ones I use, I'll be honest here, I don't use very many. I'm sure I'm guilty here and there. When I'm wrong, however, it's likely to be because I got a fact wrong, not because my logic doesn't track.
