98 regal gs no running lights

At this point...replace the entire dash cluster. Either that...or the wiring from the fuse to the dash are gonzo or has a short.

There should be a fuse marked cluster or gauges that could be blown.
The turn signals are on a separate circuit.
You could also have a problem with the wiring behind the cluster or the cluster itself.
A mechanic would check the fuse and then check for power and ground at the cluster. The gauges are grounded thru their sending units.

This is typically related to faulty or disconnected resistors located on the circuit board of the instrument cluster as the odometer does not contain removable bulbs.(it is LED) There are a few how to videos on youtube to replace the bad resistors, and it seems to be the best solution for most who have this problem. The resistors cost around .99 cents for a 5 pack and is sure to save you hundreds of dollars if you can re-solder them them yourself. Here is a link to 1 of the how to vids on youtube for 1997-2004 Buick Regal: http://www.youtube.com/watch?v=84FAjLxoK00

It may not be a bulb, as the instrument cluster usually has a number of bulbs and they probably didn't all blow at once. Pulling instrument clusters can be touchy and may not be for the average "shadetree mechanic" however, typically there are a few screws above the cluster going into the dash pad and a few hidden behind the trim panels below the cluster.

You have to remove the dash and re-solder some resisters that go bad. I did it for my 99 Regal GS and it worked perfectly. Not that hard. Saw a youtube video

The bulbs are out in the cluster.

Soldering the new bulbs in may have done something to the LH DIMMER MODULE ! The 10amp INT LPS fuse located in the I/P fuse box is good ? Is the dimmer switch at full bright ?
When the park lamps are on, battery positive voltage (B+) is supplied through the instrument panel (IP) fuse block INT LPS fuse via CKT 32 (YEL) to the dimming switch. The dimming switch supplies a variably controlled voltage to the dimmer module(s). The potentiometer in the panel dimming control cannot provide enough current in order to operate the panel lamps. The dimmer module(s) amplifies the current, supplying the variable voltage to the panel lights. As the amount of the current increases, the lamps become brighter. As the amount of the current decreases, the lamps become dimmer.
Buick
There is only 1 dimmer module that supplies current to the panel lights. The left dimmer module supplies current via CKT 8 (GRY) to the following components:
• The headlamp switch
• The door switch flood lamps
• The heater and A/C control
• The door handle flood lamps
• The passenger temperature control
• The ashtray lamp
• The shift select switch lamp
• The instrument cluster
• The steering wheel controls
• The reset switch
• The trip odometer
• The radio
Parade Dimming
The parade dimming feature prevents the dimming of the VF displays when daylight is sensed by the lamp control module (LCM) and the park lamps are turned on manually. If the LCM senses night time conditions and the park lamps are on, the LCM allows the VF displays to be dimmed.
All of the modules with the VF displays receive the dimming level from an input on CKT 8 (GRY) or CKT 1244 (BRN). When the LCM determines that the above conditions are met, the LCM sends a signal to the instrument cluster and the radio via CKT 390 (YEL). For Buick, the radio then relays this information to the heater and A/C programmer and passenger temperature control via CKT 724 (PPL/WHT). The modules with the VF displays then adjust the dimming level of their VF displays to match the level of the incandescent lights.
LH Dimmer Module
Under the left side of the IP, on the left side of the steering column brace

check the alternator charge level .
check the battery + and - are not lossing
if nothing then you have wiring short

I have ran into this problem myself. I had to get a new cluster from the dealer and have them reprogram the old mileage into the new cluster. I have seen this problem a lot in ford.

sounds like a faulty headlight switch w/dimmer