In the space of distributions, how big is the subspace of functions?

Question

I'm teaching Distribution theory and many of my students still believes that there is only one kind of distribution : the distribution that can be identified to a $L^1_{\text{loc}}$ function. And I know that there is plenty of examples of distributions that can not be represented with a $L^1_{\text{loc}}$ function. But when I say plenty, can we say how big the subspace of distributions that comes from a $L^1_{\text{loc}}$ function is in the space of all distributions ? I don't know if there is a way to mathematically measure this, if not, just tell me that there is no easy way to compare this two spaces and why.