For a smooth projective hypersurface $H \subseteq \mathbb{P}^n$ of degree $d$ one can calculate its Kodaira dimension $\kappa(H)$, and find $$\kappa(H) = \begin{cases} -\infty \qquad &\mbox{if } d < n +1,\\ 0 &\mbox{if } d = n+1,\\ \dim H &\mbox{if } d > n+1. \end{cases}$$

What about if $H$ is not smooth? Can we say anything about its Kodaira dimension, or even sensibly calculate something we can call the ''Kodaira dimension''?

I've only ever seen canonical divisors defined over smooth varieties, so am unsure how to proceed in the singular case.

canonicalsingularities you can compute the Kodaira dimension the same way as in the smooth case. For the definition of the canonical sheaf/divisor see mathoverflow.net/questions/35736/…. $\endgroup$ – Sándor Kovács Mar 4 at 15:54