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Are you in a context where you use vector norms with several $p$-values? Then write them as $||x||_p$.

If you only use the one with $p=2$, and no other confusing notation, then it is quite safe to omit the subscript.

If you instead are interested in the question of which is used more often, it is difficult to know in general. Maybe some text mining approach at Arxiv could answer? In the meantime you could take a look at some random or "random" Arxiv papers and see what is done there.