Examples of non Quillen-equivalent model categories having equivalent homotopy categories I am looking for examples (references) of pairs of non Quillen-equivalent model categories having the same homotopy categories. 
The motivation is of course that I have two model categories and all the attempts to prove that they are Quillen equivalent have failed so far, but the homotopy categories are equivalent as category. I cannot find in this case any zig-zag of adjunctions between them. Reading such examples could help me to understand better the problem.
 A: The ones which usually come up to my mind as soon as I think of this:


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*The categories of modules over $\mathbb Z/p^2$ and $\mathbb F_p[\epsilon]/(\epsilon^2)$, $p$ a prime integer. These rings are quasi-Frobenius, so their module categories have a model structure where cofibrations are monomorphisms, fibrations are epimorphisms and weak equivalences are homomorphisms which become isomorphisms in the stable module category, obtained from the module category by killing projective-injective objects. The homotopy category is this stable module category, which in both cases is the category of $\mathbb F_p$-vector spaces. This example is equivalent to Rasmus'.

*The category of DG-modules over $\mathbb F_p[v_n^{\pm1}]$, $|v_n|=2p^n-2$, $d(v_n)=0$, and the category of modules over Morava's $K(n)$.

*The model category of spectra localized at $K_{(p)}$-equivalences, where $K$ is complex $K$-theory and $p$ is an odd prime, and Franke's algebraic model category $C^{(T,N)}(\mathcal A)$ defined in http://www.math.uiuc.edu/K-theory/0139/. This doesn't happen for $p=2$ by results of Constanze Roitzheim.
In all these cases the model categories are stable and the homotopy categories are not only equivalent as categories but as triangulated categories.
A: Daniel Dugger and Brooke Shipley give an example in their paper
A curious example of triangulated-equivalent
model categories which are not Quillen equivalent
available here.
