In Heath, Lutzer, Zenor, Monotonically Normal Spaces, Trans. Am. Math. Soc., 178, 481-493, (1973), the authors discuss an example of a countable regular space which is not monotonically normal (see Example 7.3 on pg. 490). The example is of some interest as it immediately indicates the existence of a countable regular space which is not stratifiable.
No construction of the space is supplied, and instead a reference is given to the first named author's earlier paper An easier proof that a certain countable space is not stratifiable, which appears in conference proceedings that I could not find in print.
Does anyone know the construction of the space discussed above?
I would also be happy to see any other construction of a countable regular space which is not monotomically normal (or equivalently not stratifiable). Indeed, another example of such a space was supposedly constructed by van Douwen (see Nonstratifiable regular quotients of separable stratifiable spaces, Proc. Amer. Math. Soc. 52 (1975), 457–460) although again references appear in lieu of details. This time to a pair of the author's preprints which may have never actually seen the light of day.
Completely revised.
In The cometrizability of generalized metric spaces, Section $4$, Taras Banakh and Yaryna Stelmakh construct a regular topology $\tau$ of weight $\omega_1$ on $\Bbb Q$ such that $\langle\Bbb Q,\tau\rangle$ is not cometrizable and hence not stratifiable. (I have not yet gone through the construction.)
I had not previously encountered cometrizability; $X$ is cometrizable if $X$ admits a weaker metrizable topology such that each point of $X$ has a (not necessarily open) nbhd base consisting of sets that are closed in the metrizable topology. Apparently P.M. Gartside and E.A. Reznichenko proved in Near metric properties of function spaces, Fund. Math. 164:2 (2000), 97–114, that every stratifiable space is cometrizable.