Is the expression 'What's one say?' corrent?

Solution 1:

From English Pronunciation in Use (Martin Hewings, 2007)...

When does follows a wh- word¹, it can be pronounced /s/ or /z/, but isn't left out completely.

What does he do? = What's he do?
When does it start? = When's it start?
(not What he do? When it start?)

As per @Greg's comment, this particular contraction is almost never used in other contexts. Also, it won't always sound "natural", and unlike more "standard" contractions (can't, isn't, won't) there's rarely any reason to transcribe it, and your speech won't ever sound stilted even if you never say it.

¹@Janus Bahs Jacquet's comment below is highly relevant...

Note that wh- word here doesn’t necessarily just mean the interrogative word itself, but the entire interrogative phrase it’s in; thus...
What time’s it start?
Who in the bloody hell’s he think he is?
...both work fine as well.