Most European cards use Chip & PIN with pin preference: If the terminal supports it the card will ask for a chip instead of a signature. Chip-based signature transactions are only used as fallback if the terminal doesn't support pins (which is very uncommon).

Unfortunately most US cards don't use a pin code at all. There are a few issuers that provide you with a pin code, but even in those cases there's typically a signature preference and the pin code is only used as fallback if the terminal doesn't support signatures. So if you're using a US card in Europe pretty much the only places where you can use the pin code are automated terminals (e.g., at train stations). At most other places you're going to have to use a signature.

There are a few US issuers that issue Chip & PIN cards with PIN preference - for example UNFCU and First Tech Credit Union. Those cards work great in Europe, but are sometimes a pain to use in the US. E.g., if you're paying at a restaurant and you have to go with the waiter to the terminal in some backroom, because the terminal asks for the pin. That's not an issue in Europe, as waiters usually carry mobile terminals with them.