PROBLEM: You're getting really annoyed by that chirping on every pedal stroke.
SOLUTION: Silicon spray is the best fast fix we've found.
Check that your cleats are on tight and not terribly worn. Make sure there's no mud or gravel stuck in them. Check for stuff in the pedals, too.
Everything look OK? Spritz silicone around the edges of the cleats and on the pedal surface. Think about where there's contact and make it wet.
Why silicone? Because it's slippery but not sticky. There's no gunky buildup. Unlike other types of lubes, it won't pick up stuff that you walk on after you apply it.
This fix will stop the squeaks -- for a while. Silicone is light and not long lasting, so keep a can where you park your bike. You'll need to use it every few rides.
Think about replacing your cleats. Squeaking usually means they're getting pretty worn.
Quick Tip: If you always put the same foot down for traffic stops, that cleat will wear faster. So occasionally switch the cleats (or the foot you put down) to extend their life.
Before switching cleats, outline them with a pencil or marker. Then you can reattach them in exactly the right place.