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We discuss some recent results on singularity properties of word maps and Lie algebra word maps on semisimple algebraic groups and semisimple Lie algebras, generalizing a work of Aizenbud-Avni done in the case of the commutator map. These singularity properties have a deep connection to asymptotic point count over finite rings of the form Z/p^kZ. We utilize this connection to extract information on certain natural random walks induced by these word maps on compact p-adic groups, providing applications to p-adic probabilistic Waring-type problems.

Based on a joint work with Yotam Hendel.