We recently got a new stainless steel pan. We’ve been using it on the burners of our gas stove, on a regular basis. Small, pale white spots have appeared on the bottom of the inside of the pan…we live in the Los Angeles area (about 60 years now). What’s the best way to get the spots off?
I’m also in the LA area. Sounds like hard water deposits, try boiling white vinegar (or any vinegar, but don’t waste the good stuff) diluted in water. Don’t let it boil dry. Dump the boiling vinegar water after a few minutes and try a scrub with a scouring pad if the spots are still there.
Most likely, the spots are mineral deposits. If so, Lime-A-Way or CLR should banish the spots. These products contain a strong acid and you should wear rubber gloves and eye protection when using them.
Or try a milder acid like vinegar first 
Or Bar Keepers Friend, which you should be keeping around anyway.
Not surprisingly, it is also an acid (oxalic).
I’ll second Bar Keeper’s Friend. Works great on my stainless brew pots.
Try real hard to avoid scratching the surface. Any scratches will really attract mineral deposits going forward. And food gunk. And make food stick more easily.