With the ambiguity that tends to accompany separation axioms, I think it prudent to define my terms first:
A regular space is one in which a closed set and a point not contained in it can be separated by open neighborhoods.
A normal space is one in which disjoint closed sets can be separated by open neighborhoods.
The following examples come from $\pi$-Base, which is a searchable database of spaces from Steen and Seebach's Counterexamples in Topology.
The following spaces are regular but not normal. You can learn more about them by viewing the search result.
$[0, \Omega) \times I^I$
Deleted Tychonoff Corkscrew
Deleted Tychonoff Plank
Dieudonne Plank
Hewitt’s Condensed Corkscrew
Michael’s Product Topology
Niemytzki’s Tangent Disc Topology
Rational Sequence Topology
Sorgenfrey’s Half-Open Square Topology
Thomas’s Corkscrew
Thomas’s Plank
Tychonoff Corkscrew
Uncountable Products of $\mathbb{Z}^+$
The following spaces are normal but not regular. You can learn more about them by viewing the search result.
Countable Excluded Point Topology
Divisor Topology
Either-Or Topology
Finite Excluded Point Topology
Hjalmar Ekdal Topology
Nested Interval Topology
Right Order Topology on $\mathbb{R}$
The Integer Broom
Uncountable Excluded Point Topology