Do Hausdorff spaces that aren't completely regular appear in practice? Completely regular spaces include all metrizable spaces, topological vector spaces, and topological groups in general. In fact, they are exactly the uniformizable spaces.
Complete regularity is hereditary, ie. a subspace of a completely regular space is also completely regular, and it's preserved by arbitrary products. The completely regular spaces are in fact a reflective subcategory of $\mathrm{Top}$, as can be seen from another characterization: they are exactly the spaces for which the real functions determine the topology, ie. the topology is the initial topology for the set of real continuous functions.
Finally, every subspace of a compact Hausdorff space is completely regular, and conversely, every ($T_0$) completely regular space embeds universally into a compact Hausdorff space via the Stone-Čech compactification.
There are of course many natural examples of topologies that aren't completely regular. The two that I know of are Zariski topology, which is $T_1$, but not Hausdorff, and Alexandrov topology, which is a natural topology on a poset that can't even be $T_1$ in an interesting way.
What I haven't seen before are examples of Hausdorff topologies that aren't completely regular, and weren't constructed specifically for the purpose of being a counterexample. Considering how widely topology is applied (and how little I know of it) I'm assuming there are some, and I'd be interested in hearing how often they appear. 
The strengthening of this question to normal spaces seems to have elementary and satisfactory answers: the topology of pointwise convergence on $\mathbb R$ isn't normal (witnessing the fact that normal spaces aren't closed under products).
 A: Hint:  This  is a  starter which might enable  research in   the net. In the classic Counterexamples in  Topology   by  L.A.    Steen and  J.A.
Seebach,  Jr. there is a general reference chart  in the appendix  listing  $61$ properties of  a  total of  $143$ topological spaces.
The following topological spaces have the wanted properties:

T2 spaces which are  not completely regular:

*

*(47): An altered long line


*(60): Relatively prime integer topology


*(61): Prime integer topology


*(63): Countable complement extension topology


*(64): Smirnov's deleted sequence topology


*(66): Indiscrete rational extension of $\mathbb{R}$


*(67): Indiscrete irrational extension of $\mathbb{R}$


*(68): Pointed rational extension of $\mathbb{R}$


*(69): Pointed irrational extension of $\mathbb{R}$


*(72): Rational extension in the plane


*(74): Double origin topology


*(75): Irrational slope topology


*(76): Deleted diameter topology


*(77): Deleted radius topology


*(78): Half-disk topology


*(79): Irregular lattice topology


*(80):  Arens square


*(81):  Simplified Arens square


*(88):  Alexandroff plank


*(90): Tychonoff corkscrew


*(91): Deleted Tychonoff corkscrew


*(92):  Hewitt's condensed corkscrew


*(94): Thomas's plank


*(96): Thomas's corkscrew


*(100):  Minimal Hausdorff topology


*(113): Strong ultrafilter topology


*(125): Gustin's sequence space


*(126):  Roy's lattice space


*(127): Roy's lattice subspace


*(133): Tangora's connected space

