We have a structure theorem for f.g. modules over $R$, whenever $R$ is a PID. In the case of $R=\mathbb{Z}/p\mathbb{Z}[x]$ ($p$ a prime), the structure theorem can be used to obtain the rational canonical form for matrices over the finite field $\mathbb{Z}/p\mathbb{Z}$.



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> I am interested in some kind of canonical form for matrices over $\mathbb{Z}/p^k\mathbb{Z}$. Is there such a canonical form in the literature?
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> This question naturally leads to a more concrete one: What is known about the structure of f.g. modules over $\mathbb{Z}/p^k\mathbb{Z}[x]$?
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If I have a matrix $A \in \mathrm{Mat}\_n(\mathbb{Z}/p^k\mathbb{Z})$, the relevant $\mathbb{Z}/p^k\mathbb{Z}[x]$-module is the ``vector space'' $(\mathbb{Z}/p^k\mathbb{Z})^n$, on which $x$ acts as multiplication by $A$.



Rational Canonical Form over $\mathbb{Z}/p^k\mathbb{Z}$
The problem is open, and not because nobody tried. For instance, it is known that the number of similarity classes in $M\_n(\mathbf Z/p^2 \mathbf Z)$ is equal to the number of simultaneous conjugacy classes of pairs of *commuting* matrices in $M\_n(\mathbf Z/p\mathbf Z)$ (S. J AMBOR and W. PLESKEN , *Normal forms for matrices over uniserial rings of length two*, J. Algebra 358 (2012), 250–256). For small values of $n$, exact formulae are given in my paper in IUMJ *[Similarity of matrices over local rings of length two](http://dx.doi.org/10.1512/iumj.2015.64.5500)* with Pooja Singla and Steven Spallone, which is also available on the [arXiv](http://arxiv.org/abs/1212.6157).


From the classification/structure theorem point of view, these are *wild problems*, so you should not expect a good answer. You will find a discussion of the history and other results on this problem in the intro to my paper mentioned above.


