It's also up to the individual brokers. Some have a LOT of exotic pairs, and some have only the majors.
One thing I have noticed. If you have a European broker, the spreads on exotic (or not so exotic) EUR/xxx pairs are often lower than with a US broker. I really should check to see if a Japan-based broker offers lower spreads on some of the more interesting JPY crosses.
You can trade anything your broker offers, but Nancy is correct that exotics have higher spreads (sometimes incredibly high spreads) and can also be very volatile. They also have lower liquidity which can lead to higher slippage or requotes entering and exiting the trade. Still, some people do trade them profitably, but it's definitely not the same as trading the EUR/USD or GBP/USD.