I had a scary no-start condition: the headlights and accessories came on, but the starter did not even click. The battery and check engine light were on.

Turns out it was because my shift into Park somehow did not engage correctly. The shifter appeared to be in Park, but it wasn't. After wiggling the shifter around a little, it started right up.

Mainly posting this for future searches. But for discussion's sake, it is a shame that the LS can't use its nice text display (for range, MPG, low tire alert, etc.) to explain what's going on.

 

There is a button next to the shift lever that lets you move the lever itself without changing the "mode".
You may have caught that at some point, or the mechanism that releases the lever from the gearbox linkage may be worn.
Whenever I have a "will not start" condition I ALWAYS cycle the gear lever away from and back into Park and try again. 99% of the time it fixes the problem.
You can also start the car in NEUTRAL so worth trying that as well.

I had similar issues with both my Acura RLs and with my S-Class so it seems to be endemic to automatic gearboxes.