It depends on my current outfit, but usually I wear hats.
Hats can be pretty frustrating sometimes though. My favourite hats like the Straw Hat block me from getting tanned, and I like to keep my character with black/dark skin. I have to constantly take it off when I go to the island so she gets a tan. It sounds petty and lazy, but thats just me :I
I always wear headgear, but always stuff that shows off my hair too. My mayor wears a small crown on his spiky hair (so there's a spike going through the crown, even), and the other residents wear a flower bobber or halo, a small yellow hat worn to the side to be replaced by a small silk top hat when spring arrives, and a blue bow.
My mayor's hair is in a bun, so it looks a little funny if I wear a hat. Usually, I'll wear wigs or small accessories, but right now I'm obsessed with the Black Veil.
My favorite "hat" is the bulb bopper(sometimes I switch it with the heart bopper), it's best of both worlds. Not only does it light up during the night, but you get to see your hair too
It all hinges on my current outfit (and occasionally if I just happen to strongly feel one way or the other on any given day). If I happen to own a hat that looks fantastic with the rest of what I'm wearing, then hell yeah I'm gonna rock it. But, if I don't have a complementary hat, or if my hair seems to work particularly well, then I'm fine with going without one.
If I goof on getting my haircut and end up with a tragedy on my head, I'll cover it up as best as I can though.