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There appears to be some infinities in classical electromagnetism:

The total energy contained in the electric field of a point charge diverges.
The potential at a point charge diverges.
The potential at a line charge diverges.
The potential at a volume charge does not diverge.

Are there any references/discussions providing rigorous arguments to show these, and other similar phenomena?

I found a few rather informal explanations:

Curvature argument

Green's function

From StackExchange:

The problem of infinite energy of electron as point charge?

Electromagnetism for mathematicians

Mathematical book on "Maxwell" equation

Cheng
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