The reason why these cryptographically secure hashes are chosen is because they provide randomized output. You might find ways to calculate the best input to calculate a number of zeros for non-cryptographically secure hashes such as CRC's or hashes used for hash tables (a software methodology for storing e.g. sets of elements).
In principle you could find that a hash $h$ generated by hash function $H$ is still cryptographically secure if $H'$ derives $0 \| h$ but generally the output of cryptographic hashes have a random distribution. If that wasn't the case then the collision resistance would be lower than approximately half of the output size.
In other words, it would violate the "desirable property" that:
Non-correlation (correlation freeness): hash function inputs and outputs should not be statistically correlated; that is, even a small change in the input should drastically affect the output bits; this phenomenon is called the avalanche effect.
Quoted from the paper "Cryptographic Hash Functions: Recent Design Trends and Security Notions" by Saif Al-Kuwari, James H. Davenport, Russell J. Bradford.
The way that modern hashes are created you can assume that this property holds; it certainly does for SHA-1, SHA-2 and SHA-3 and most other hashes based on bit operations / symmetric cipher techniques.