Can somebody point me to examples of distributions which are not sums of delta functions or derivatives thereof. Also of course not integrals with local integrable functions.
Additional: is there a classification?
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Joelafrite made a good suggestion: consider the first distributional derivative of a function $f$ that is not absolutely continuous. By definition, this distribution acts as $$\phi\to -\int f\phi'$$
And of course, higher distributional derivatives of the same functions provide even more singular examples.
As for classification: a derivative is of order at most $k$ if its value on test function $\phi$ is bounded by $C^k$ norm of $\phi$. So, order $0$ distributions are signed measures. There are distributions of every finite order, as well as of infinite order.