Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So:
The intricacies of arithmetic at singular cardinals notwithstanding, I think you're looking for something which doesn't exist.
Given any cardinals $\kappa,\lambda$, singular or regular, there is a (set) forcing extension preserving cardinals and cofinalities in which $2^\kappa>\lambda$. Upper bounds only appear in a meaningful sense when we try to control the value of the continuum function on many inputs simultaneously. For example, famously Shelah proved that $2^{\aleph_\omega}<\aleph_{\omega_4}$ provided that $2^{\aleph_n}<\aleph_\omega$ for all $n<\omega$, but it's perfectly consistent that $2^{\aleph_\omega}\ge\aleph_{\omega_4}$ since for example we could have $2^{\aleph_0}=\aleph_{\omega_{17}+42}$ already.
