What are the simplest/ typical examples of a Serre fibration which is not a Hurewicz fibration? Is it something pathological?

Sorry if the question is too elementary for MO.

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A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space are both CW complexes is necessarily a Hurewicz fibration. M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577. (The proof is corrected in R. Cauty. Sur les overts des CW-complexes et les fibr\'es de Serre. Colloquy Math. 63(1992), 1--7). No relationship between the covering map and the CW structures is required. This argues either that counterexamples are pathological or that it is a special property for the total space of a Serre fibration with CW base space to be a CW complex, although it has the homotopy type of a CW complex if the fiber does.

These show up all the time in various generalized covering space theories, the reason being that you want homotopy lifting with respect to a certain class of spaces that you're interested in (e.g. locally path-connected spaces) which includes all cubes $[0,1]^n$ but probably not all spaces.

An important example with a simple construction is the generalized universal covering $p:\widetilde{\mathbb{H}}\to\mathbb{H}$ of the Hawaiian earring $\mathbb{H}$ that I describe in this answer. It is characterized by it's lifting property (described in link) which shows it to be a Serre fibration. It is not a Hurewicz fibration since it doesn't have homotopy lifting with respect to non-locally path-connected (even contractible) spaces like $\displaystyle\frac{[0,1]\times\{1,1/2,1/3,...,0\}}{\{0\}\times\{1,1/2,1/3,...,0\}}$.

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