Hats for men

Not taboo. It might identify you as an American, unless it is a New York Yankees cap, you're 25 years old, have a zillion tattoos, and several piercings. Then you'd look like a typical European young person.

Taboo? You won't be asked to leave the country...

Stylish? No

Common? No

But if you need to protect your head from the sun, by all means wear a hat.

Baseball caps are pretty rare in Europe. You will stand out (as stated in other posts.) I don't see lots of European men wearing "safari" hats....IMHO that will also mark you as a tourist just like a baseball cap will. However, there are lots of types of hats. I wear a broad brimmed hat for maximum sun protection (just like I do when I walk or hike) but I don't mind looking like a tourist. My attitude, manners and friendliness are more important to me than dressing like a native because I can't pull that off anyway. A Fedora, Greek Fisherman or other hat will blend in better with the European sense of style. So, if you are trying to go with the flow these are probably your choices..... wear a hat you don't think is stylish (the small Irish/Italian caps) or swing by a hat store and pick something you like.

What marks out American clothing and hats from European is, to a large degree, the advertising on them.

Unless, as in the prior example, they are patrons of certain chain stores buying items for some sort of fashion statement, most Europeans prefer shirts, jackets, hats, etc., without printed advertising.

Some people occasionally wear caps, but they are usually one solid colour - usually not bright - with maybe a contrasting thread around the brim.

Many people wear wide brimmed hats - like cricket hats or wide brimmed walking hats. Some wear straw hats. Quite a few keep a wax hat for the weather.

And all bets are off at sporting activities where Dutch people will be easily told apart by all the orange clothing from the head down, and supporters of particular teams wear uniforms or jackets and hats in the team colours and sponsor names, and in motor racing on 2 or 4 wheels the number of their favourites.

But turn up at a museum or walking around town or on a trail with a gimme cap we'll spot you a mile away.

Does it matter? Not a jot.

Are baseball hats taboo in Europe?

No. They are sold widely, and worn often by the locals. If you are concerned they may label you as an American, wear something like this:

http://store.manutd.com/stores/manutd/products/product_details.aspx?pid=147067&cid=48960

and abracadabra you are now European!

I almost always wear a baseball cap to protect my bald head :) I wore one all over Paris and London a couple of years ago. And last year in Italy, too. Does it make me look "American"? I guess. Did "Europeans" notice or care? No. I took it off in churches and other places where it seemed appropriate to do so and nobody blinked. The hat I wore to Italy in 2013 was a Boston Red Sox cap. We left the day they captured one of the guys allegedly responsible for bombing the Boston marathon. Going through the passport check in Frankfurt, the agent looked at my hat and said "Boston Red Sox?" and gave me a thumbs up. Walking in Rome, near the Vatican, a random person on the sttreet yelled out "Go, Sox!" Wear your baseball hat if you want to. Be respectful and remove it in churches and other places where it seems appropriate. If you do, nobody will notice or care.

I like the Tilley airflow myself - a broad rimmed hat with a bit of mesh around the top rim for cooling. I am also concerned about sun exposure. The hat packs flat, and (hand) washes like a charm - I hang it out overnight and it's dry by morning. Made in Canada, but the comments I've gotten aren't that I'm Canadian, but rather "Cowboy!" I were it at home and on travel.

Probably nobody but you, and possibly your travel companion, will notice what you wear.
So wear what you feel comfortable with, would be my thought.