Suppose two signature schemes provides the same level of security. The sum of the sizes of a public key and a signature, i.e., sizeof(sig) + sizeof(pk), are equal in the two schemes. One of the schemes has smaller public key size, and the other has smaller signature size. Then which scheme is preferred in practice? If it depends, then in what scenarios we prefer smaller public keys, in what scenarios we prefer smaller signatures?
Here is a more specific example. Supppose we want to use a multi-party signature scheme. We have two choices, both providing $256$-bit security: 1. A scheme where a group of $N$ users has a shared public key of size $512$ bits and produces signatures of size $512N$ bits; 2. A scheme where a group of $N$ users has a shared public key of size $512N$ bits and produces signatures of size $512$ bits. We don't have a both-optimal scheme and have to make a trade-off.