Here's a fairly standard one (it's an exercise in Hartshorne). In an integral domain $R$ of finite type over a field, every maximal ideal has the same height (in particular, every closed point has the same dimension). Indeed, it would be natural to define a ring to be equidimensional if every maximal ideal has the same height. Here's a problem with this definition.

Suppose now that $R$ is a DVR with parameter $r$. Consider the ring $R[x]$. This ring has one maximal ideal of height one, $\langle xr - 1 \rangle$, and another maximal ideal of height two, $\langle x, r \rangle$.

The point being, this is a domain, so its $\text{Spec}$ is presumably equidimensional, of dimension 2 the Krull dimension of $R[x]$. But it has closed points of different heights (although with very different residue fields). Of course, this isn't as pathological as a non-catenary ring, but we can even assume that $R[x]$ is a localization of $k[r,x]$.