I live in AZ and recently had a water line break. It is in the process of being fixed, but apparently we leaked $3,000 worth of water! The water line was NOT very far under the ground and was in fact less than a foot away from the surface for the first few feet. Does anyone know how far underground the water line needs/is supposed to be that runs out front from the water meter to my house?

In most localities utilities have to be below the frost line for that area... due to to frost heaving and soild movement (rocks get moved too and act like saws over time).

I think that may be differing in regulations between the northern part of the state that sees colder weather and snow than it is in the southers much hotter areas.

I assume you are in the southern hotter dryer part of the state.

This study discusses the frost depth but it is your local building department that has the last word, call them, its free. Maximum frost depth and freeze-thaw frequency measurements and simulations at Bellemont, Arizona | PYRN (http://pyrn.ways.org/maximum-frost-depth-and-freeze-thaw-frequency-measurements-and-simulations-bellemont-arizona)

It was in fact less than a foot away from the surface for the first few feet. And how deep did it go after leaving the meter? That's how you tell.
You ask about depth. Different areas have different temperatures and frost lines. For instance, I live in Florida where the water lines are mandated at least 18" below grade but back in Wisconsin our water and sewer ran 6 feet beneath grade. It all depends upon where you live and how cold it can get. To learn more contact your local Building department.
Regards, Tom