Rank depending on a parameter

Determinant. Adding all rows to the first and factoring:
$$\det A_m = (m+2)(m-1)^2.$$

Case $m \neq -2$ and $m \neq 1$. $\det A_m \neq 0$, so $\operatorname{rank}(A_m) = 3$.

Case $m = 1$. All entries are $1$, so every row is equal. The rank is $1$.

Case $m = -2$.
$$A_{-2} = \begin{pmatrix} 1 & 1 & -2 \\ 1 & -2 & 1 \\ -2 & 1 & 1 \end{pmatrix}.$$
The sum of the three rows is $0$, so the rows are linearly dependent. The first two rows are not proportional, so the rank is $2$.

Conclusion.
$$\operatorname{rank}(A_m) = \begin{cases} 3 & \text{if } m \neq -2 \text{ and } m \neq 1, \\ 2 & \text{if } m = -2, \\ 1 & \text{if } m = 1. \end{cases}$$