Why do we keep designing new static s boxes? As I understand s box properties, they primarily need high non linearity and low auto correlation.  But most importantly for this question, they're not secret.

There are many questions on this site asking for help in generating them.  And new ciphers always seem to have new s boxes.  Why?  There are proven numerical optimisation techniques such as tree search and hill climbing to optimise a function.  Could we not just devote an hour of a large computer cluster to generate optimum (say) 3, 4, 5, 6, 7 and 8 bit wide s boxes?  Universities have access to such resources.  They could then be published in a book rather like they used to publish random numbers.

Note: By static s box, I mean one of those that isn't related to a key like Blowfish's.
There are simply a long list of properties for S-boxes that are relevant, see for example these questions:


Desirable S-box properties
S-box design criteria and random sboxes


In addition people have been trying to design S-boxes that are fast to implement in hardware or in software or which can be masked efficiently. For example the Midori S-box is designed to be especially energy efficient.

The search space for S-boxes is actually quite big as well; for 4-bit S-boxes (pretty small) there are $16! \approx 2^{44}$ possibilities.

This might be a nice read: Cryptographic Analysis of All 4×4-Bit S-Boxes