How many zeros must be in a bitcoin share for a pool to accept it? In the binary representation of the hash.
Now the hash should be about 80 zeros.
In the share - less, but how much exactly?
At the dawn of bitcoin in a share, it was enough to have 32 zeros. Now the difficulty has greatly increased.

What are the share requirements for popular pools?
First of all, Bitcoin does not actually use the number of zeroes for the difficulty. Rather it interprets the hash as a number and checks that the block hash is less than or equal to a particular target. This makes the hash have several leading zeroes, but those zeroes are not being considered directly.

The target for a share varies by pool, and usually, by miner as well. Miners can choose a target that matches their hash rate so that they aren't submitting shares too quickly or too slowly. Of course the shares are weighted by difficulty (i.e. target). So a share with a higher difficulty (lower target) is worth more than a share with a lower difficulty (higher target).

Most pools offer setting the difficulty to many values. For example, Slushpool lets you set it anywhere between 128 and 500,000.