What's the best/safest stuff to use to clean the original, non-sagging headliner in my car? I don't want to create any unnecessary problems for myself. Thanks.
Too late, you said "headliner," so it is going to sag now. Is it cloth or vinyl? For vinyl I've used Saddle Soap. Formula 409 or Fantastic should work as well. I don't know if this would help, but instead of spraying the cleaner on the headliner, try spraying it on a cloth, then rubbing down the headliner. That should limit how much soaks into it. Just don't rub too hard.
Saddle Soap, Yes. Ditto on the saddle soap. I've used saddle soap and warm/hot water with a sponge and a little gentle wiping. Don't scrub or push, the mixture will clean the surface of loose dirt and dust. Do small sections at one time, wiping down each section with a clean towel. Don't spray any cleansers on the headliner! Good luck.
Like Pete said, don't scrub. If it's really bad you can clean it lightly, wait a week, do it again, etc. Eventually it'll all come off.
That will be a bit more difficult, but still try what we said. Stay away from cleaners with solvents (ammonia, alcohol, etc.). Use something soap-based. Don't saturate the headliner. You could try a small brush - but do it very lightly, just trying to work out the dirt, not scrub it out.