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The short answer is that over local PIRs which are not fields, all the conjugacy classes are known explicitly for $\mathrm{SL}_2$, $\mathrm{GL}_2$, and $\mathrm{GL}_3$, some can be explicitly described for $\mathrm{GL}_n$, and in general the problem is hopelessly complex (in the sense that it is wild).

The longer answer is as follows. The cases $\mathrm{SL}_2$ and $\mathrm{GL}_2$ over finite local PIRs have been known for a long time, and can easily be worked out directly. The case of $\mathrm{GL}_2$ and $\mathrm{GL}_3$ over arbitrary local PIRs has been treated more recently by Avni, Onn, Prasad and Vaserstein in Similarity classes of 3x3 matrices over a local principal ideal ring.

Trying to tackle the $\mathrm{GL}_4$ case, one runs into the problem of classifying pairs of $2\times2$ matrices over the residue field, under simultaneous conjugation. More generally, the problem of describing the conjugacy classes in $\mathrm{GL}(4n)$, \mathrm{GL}_{4n}$, for all$n$contains the matrix pair problem over the residue field. This is a wild classification problem (see the above link) and hence one cannot expect a useful explicit classification in general. I think (but I have not checked it carefully) that if you do not assume that the ring is a PIR, you run into the matrix pair problem already for the groups $\mathrm{GL}(2n)$.\mathrm{GL}_{2n}$.

On the other hand, despite the lack of a general canonical form for matrices over local rings, there is a canonical form for a large subset of matrices, notably the regular ones. Regular elements were defined by Hill (for matrices over local PIRs) in Regular Elements and Regular Characters of $\mathrm{GL}_n(\mathcal{O})$, and are characterised by the property of being conjugate to their respective companion matrix, i.e., the canonical form consists of a single companion matrix block. One can of course generalise this to a canonical form with several companion matrix blocks. This is discussed in detail in Guralnick, Similarity of Matrices over Local Rings. Over fields this is the familiar rational canonical form.

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The short answer is that over local PIRs which are not fields, all the conjugacy classes are known explicitly for $\mathrm{SL}_2$, $\mathrm{GL}_2$, and $\mathrm{GL}_3$, some can be explicitly described for $\mathrm{GL}_n$, and in general the problem is hopelessly complex (in the sense that it is wild).

The longer answer is as follows. The cases $\mathrm{SL}_2$ and $\mathrm{GL}_2$ over finite local PIRs have been known for a long time, and can easily be worked out directly. The case of $\mathrm{GL}_2$ and $\mathrm{GL}_3$ over arbitrary local PIRs has been treated more recently by Avni, Onn, Prasad and Vaserstein in Similarity classes of 3x3 matrices over a local principal ideal ring.

Trying to tackle the $\mathrm{GL}_4$ case, one runs into the problem of classifying pairs of $2\times2$ matrices over the residue field, under simultaneous conjugation. More generally, the problem of describing the conjugacy classes in $\mathrm{GL}(4n)$, for all $n$ contains the matrix pair problem over the residue field. This is a wild classification problem (see the above link) and hence one cannot expect a useful explicit classification in general. I think (but I have not checked it carefully) that if you do not assume that the ring is a PIR, you run into the matrix pair problem already for the groups $\mathrm{GL}(2n)$.

On the other hand, despite the lack of a general canonical form for matrices over local rings, there is a canonical form for a large subset of matrices, notably the regular ones. Regular elements were defined by Hill (for matrices over local PIRs) in Regular Elements and Regular Characters of $\mathrm{GL}_n(\mathcal{O})$, and are characterised by the property of being conjugate to their respective companion matrix, i.e., the canonical form consists of a single companion matrix block. One can of course generalise this to a canonical form with several companion matrix blocks. This is discussed in detail in Guralnick, Similarity of Matrices over Local Rings. Over fields this is the familiar rational canonical form.