I thought phrases like "to comb a bedhair" or "to press down a bed hair" was quite common.
But Google tells me the opposite. It shows zero results.
If those phrases aren't common, what are their grammatical/idiomatic equivalents?
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"Bed hair" I understand to be "the appearance of your hair in the morning" But it's not a very common compound. It is strictly uncountable, as the countable meaning of "hair" could not apply (so *"a bed hair" is not possible). As comments indicate, this use is rare enough that native speakers won't recognise it out of context. It is not common enough to be found in most dictionaries.
You could say "I've got 'bed hair'" or "My son's 'bed hair' is impossible to comb". But not how I've used scare quotes, in speech there would be a stress and intonation pattern used to indicate that this is not a common expression, but you know what I mean.
So the common expression would be "comb [or comb down] your hair". And if you needed to emphasise that it was 'bed hair' you would need to explain it.
Comb down the hair that's sticking up because you lay on it last night.