Openings with 1.e4
Any old move will do, as Julius Breyer, an early proponent of of the hyper modern movement once said while annotating a game:
"After 1.e2-e4 White's game is in its last throes!"

French: 1...e6 with idea of 2...d5
Sicilian: 1...c5 to control d4.
Open Game: 1...e5.
Caro-Kann: 1...c6 planning 2...d5
Those are the main ones which are all considered sound. Common theme is simply to either attack the center on turn 1 or prepare to do so on turn 2.
Offbeat defenses
Modern/Pirc: 1...g6 or 1...d6 planning Kingside fianchetto
Scandinavian: 1...d5 planning either 2...Qxd5 or 2...Nf6 after 1.exd
Alekhine's: 1...Nf6 planning to bait White's pawns forward