What norms can we define on $L^p_{\mathrm{loc}}$ ? or What is the most commonly used norm on $L^p_{\mathrm{loc}}$. It is tempting to define $$\|f\|_{L^p_{\mathrm{loc}}}:=\sup_{K\;\text{is compact}}{\|f\|_{L^{p}(K)}}$$
But this can be infinite for $f\in L^p_{\mathrm{loc}}$.
There is no "interesting" or "useful" norm on $L^p_{loc}$. Meaning that there is no norm that induces the standard topology; no norm such that $||f_n||\to0$ if and only if $\int_K|f_n|^p\to0$ for every compact $K$.
Otoh $L^p_{loc}$ is metrizable, assuming the underlying space $X$ has an "exhaustion" by compact sets: If there exists a sequence $(K_n)$ of compact sets such that $K_n\subset K_{n+1}^o$ and $\bigcup K_n=X$ you can define $$d(f,g)=\sum 2^{-n}\frac{(\int_{K_n}|f-g|^p)^{1/p}}{1+(\int_{K_n}|f-g|^p)^{1/p}}$$and then $d(f_n,0)\to0$ if and only if $\int_K|f|^p\to0$ for every compact $K$.