I had the same problem with my 98 buick regal gs and it turned out to be the ignition switch.
http://www.automotivehelper.com/topic1054152.htm or Goole "ignition switch replacement 2001 regal"
where is temperature switch on 1996 buick regal
were can i locate and replace a starter for a 1998 buick regal?
How do you replace the morror assembly for a 2001 Buick Regal
How do you replace the fuel filter on a 2003 Buick Regal?
how to replace a radiator in a 1996 buick regal
You have to get it from a Buick dealer.
Flasher is bad. to replace it it is under the dash at the back of the steering column.
how to replace master cylinder on 1996 buick regal
Pull the turn signal stalk towards you to activate your high/low beams
My experience with a faulty ignition switch on this generation Regal include random warning lights coming on together and turning off at the same time. Eventually, the car will sometimes stall when the turn signal switch is used. This is due to the turn signal switch "bumping" the wires leading into the ignition switch. Apparently, the wire connections inside the ignition switch sometimes tend to loosen. Dealer cost runs around $350 to $400 to replace.