What could be a cause of a 96 buick regal firing but not starting fuel pump?

The fuel pump maybe bad or the ground for the pump is poor.Near the driver side firewall under the hood is a single connector with nothing on it.Connect this via a jumper wire to the + side of the battery.You should hear the fuel pump.This bypasses the fuel pump relay and supplies power directly to the pump.If it works then you maybe low on oil as there is an oil pressure switch in the engine which prevents the fuel pump from operating when you are out of oil.