If all three lock lights on the keyboard flash once, then the laptop is in fact completing POST.
Try turning on the laptop while holding down the <Fn> key. This will start up in a diagnostic mode, describe what you see, hear, and smell (j/k about the smell) when you start with this diagnostic boot.
What if you turn it on normally and plug in a monitor, then press <Fn> + <F8> to switch the graphics over? Do you see anything then?
Okay, here are the results:
The three lock lights do not flash at all on the initial boot. The LED touch-sensitive media controls at the top of the keyboard light up in a left to right pattern, then turn off. The power button and hard drive light/led turn on and I hear the computer fan spin up and the DVD drive whir for a second. Then the computer fan spins down and the hard drive light turns off, leaving a blank screen with only the power led still lit, no other signs of life.
I tried booting and holding down Fn and got a blinking number-lock light(the number 9 inside a lock). Otherwise, everything else was the same.
Any thoughts on what might be going on and whether or not I can fix it? Or is it just "deal with the Dell warranty people" time?
Thanks again Admiral!