When I google "drop litter", only results from British websites appear.
What is the typical way in America to warn against littering?
Is it simply "do not litter" or is the verb "drop" also used?
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To add to Laurel's great answer, here is a list of the situation for 'litter' in AmE and BrE:
In AmE, 'litter' can be used in both noun and verb forms when talking about trash that is freely on the ground and not in a trash receptacle.
As a noun:
"Put the litter in the trash can."
As a verb:
"You're littering if you throw your cigarette out the car window."
and the ubiquitous
It seems in BrE that using 'litter' as a verb is not done: the last two items are not used. Instead in BrE you'd normally say
"You're dropping litter..."
"Don't drop litter"
The latter, with 'drop', makes sense in AmE, but it is just not used at all because you would say "Don't litter".
No, Americans don't use "drop litter". Instead, you'll find signs that just say "no littering", such as this one:
There are also signs that say "do not litter":
Images from Recycle Reminders