I just got the S3 and I have run into this problem also. I went into the settings and turned on the snooze for my alarm and selected snoozed this morning but the alarm never went off again. The Samsung alarm app is very buggy.
Had the same issue with no snooze.
You have to tap the alarm to open it.
Tap the MORE button on the bottom of the screen.
Scroll the screen up (there are options underneath).
Next to the snooze, slide from off to ON.
Tap the SNOOZE and then more options are available for intervals and repeating.
Don't forget to save.