[axelrosen]

IB is the easiest American one to get for Europeans. It's been around for a while and plenty of respectable finance people have been using it (for personal trading).

If you travel to the US most major ones should open an account for you too. But they ban foreign IPs sometimes and unlocking can be hassle.

The only major problem with the US I'm aware of is the death tax exemption for non-residents is only 60000$, so you don't wanna die or need to structure appropriately.