I found some addresses in which the signatures (s part) start with the same bytes, is that a sign that an address has been using a weak nonce?
Not necessarily. It depends on how many bytes are the same.
What would it be, if the same s is re-used in the formula, but the r's are different?
I don't believe that it is possible to get the private key when s is repeated. The reason that a repeated R works is because R is part of the calculation for s which allows you to rearrange the formula for s so that you can compute the private key. The nonce term disappears in that formula because you know it is the same so it can be rearranged and written out.
But s is not used in any formula. It is a single calculation and I don't think a repeated s gives any more meaningful information about the nonce or the private key.