Easyjet/Skyeurope

Easyjet is very easy to book in the US, and often with stellar deals.
I flew from Athens to London back in 2006 (and plan to fly from Lyon to Venice later this month). The planes are nice, ticketing is extremely easy, but it is no frills. The only thing that was not ideal was that my plane left Athens about an hour late, which was no big deal.
I haven't done this, but I think checking bags can add up pretty fast on many of the airlines (then again, most airlines are doing this now).
Show up early enough and get better pick at seats, just like Southwest.

They're both equivalent to Southwest in the USA. Safe and reliable but don't expect any extra services...

I have read about some flights getting delayed for Skyeurope, but 99.99% of the reviews are positive. They have few flights between cities, so when a delay occurs it is generally a larger impact then if they had many flights. Also, make sure that you arrive early for them because they have strict rules about arrival times and check-in.

Both airlines are safe. They have to meet certain safety criteria to fly in the EU, so by the fact they are in business they are meeting those requirements. Skyeurope was incredibly easy to book over the internet, I even had to change my flight to include more luggage and it was easy to do (not so simple with North American airlines). I imagine Easyjet is probably quite the same, but I have not booked with them before.