soft question - What are the most overloaded words in mathematics?

Regular. To start off:

The regular representation of a group $G$ over a field $k$ is the action on $k[G]$ given by group multiplication.

A topology is regular if a closed set and a point not in that set can be separated by disjoint open sets.

A point $zeta_0$ on the boundary of a domain in $mathbb C$ is called regular if there exists a subharmonic barrier function $b(z)$ defined within $D$ near $zeta$. This may not be the standard definition but Gamelin's complex Analysis defines it as a subharmonic function $omega(z)$ on ${|z-zeta_0|<delta}cap D$ which is negative everywhere, tends to 0 at $zeta_0$, but $limsup(omega(z))<0$ as $z$ tends to any other boundary point of $D$ within distance $delta$ of $zeta_0$.

I've borrowed/paraphrased the following from the Wikipedia disambiguation page but removed a couple that either are not too relevant to pure math or qualify the "regularity" more. Feel free to put them in too.

Regular cardinal, a cardinal number that is equal to its cofinality

Regular category

Regular element, and regular sequence and regular immersion.

Regular code, an algebraic code with a uniform distribution of distances between codewords

Regular graph, a graph such that all the degrees of the vertices are equal

The regularity lemma, which has nothing to do with regular graphs

Regular polygon, and regular polyhedron

Regular prime: a prime $p$ that does not divide the class number of the $p$th cyclotomic field $mathbb Q[zeta_p]$.

Regular surface in algebraic geometry

Regularity of an elliptic operator

JS Milne's comment: A regular map is a morphism of algebraic varieties.

Regular value of a differentiable map

Regular ring (Note: this definition can be made noncommutative. A right noetherian ring R is said to be right regular if every finitely generated right R-module has finite global dimension. See Lam's Lectures in Modules and Rings, Section 5G.)

(von Neumann) Regular ring

Regular language, a language that can be accepted by a finite state machine.

Absolutely regular is a synonym for $beta$-mixing in stochastic processes.

Regular matroid, a matroid which is representable over every field. In this sense, all graphs are regular (their cycle matroids are regular), which has nothing to do with regular graphs.