Single class US airlines that a European might actually fly:
Southwest, JetBlue***, Midwest Express*.
Two-class US low-cost/low-fares that a European might actually fly:
America West**, Frontier, AirTran, Spirit.
Two-class US low-cost/low-fares that are now defunct:
Major two-class US carriers:
Alaska, American, Continental, Delta, Northwest, United, USAirways.
In the US, expect to pay GOBS of money for a first class ticket, unless you're flying MidWest Express, which serves only about 30 markets.
* Midwest Express is first-class only.
** America West is was started as a low-cost/low-fare but now operates as a major carrier.
*** JetBlue's coach class seating is more similar to European style First class seating, though they are still coach seats.
Those who fail to learn history are doomed to repeat it in summer school.