There's a thread on airliners.net about the same topic.
One of the most prominent and useful aspects of the delta wing is the fact that it increases an airplane's critical Mach number (the speed at which some air passing over any part of an aircraft - usually the wing - reaches Mach 1.0).
By increasing the critical Mach number, it makes supersonic flight much more feasible on delta wings than on conventional wings.
Delta wings are particularly happy even at a high AoA (angle of attack). This is because of the reaction lift generated by the wing, and the vortex created at the leading edge of the wing. Hence, airplanes with delta wings have a much higher tolerance to stalling. This makes it easier to attain higher vertical speed, also known as "rate of climb" safely.
But on the flip side:
One of the downsides of deltas is that they create more drag, because of the bigger surface area, but they do have a bigger internal volume (for fuel, for example) which could lead to a smaller fuselage, or more space in the fuselage for other things. Another disadvantage is higher landing and take off speeds.
The last point is particularly important. The Concorde's already small number of usable airports was reduced because the runways needed to be long enough to accommodate for the 160 KIAS landing speeds (for comparison, typical turbofan-based airliners lands at 130-140 KIAS).
As a result of the higher takeoff/landing speeds, the cost of maintenance of vital components such as the brakes and tyres is higher. Moreover, the high landing speeds can even lead to greater stress on the airframe. So, indirectly delta wings do contribute to a higher cost of maintenance.
Other disadvantages of delta wings I can think of is the lack of a flaps/slats system. Delta wings don't require flaps as they can simply reduce their airspeed by increasing the AoA, but the lack of flaps reduces their subsonic efficiency, and increases the takeoff distance and takeoff velocity. For instance, the Concorde's delta wings produced very little lift during takeoff prior to rotation, necessitating high takeoff speeds of nearly 220 knots!
As an answer to your second question, I could not find anything about regulations relating to the use of delta wings on airliners.
All in all, delta wings work well for niche purposes such as supersonic aircraft, but their disadvantages and poor operating economics make them impractical for subsonic flight.
However, as you can see here, the now-scrapped Boeing Sonic Cruiser attempted to make the best of both worlds by combining a conventional wing with a delta wing, to form a wide, "cranked-delta" wing design.