Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?

To be more detailed: if I want to show that some distribution is in fact a smooth function, I can try to use the concepts of singular support or wavefront, together with hypoellipticity. What can I do if I want much less, namely to show that a distribution is only a locally-integrable function (not necessarily smooth)?