= Solution
The <minimal polynomial> divides the annihilating polynomial $t^2(t-1)$, since $A^3-A^2=0$. Every nonconstant monic divisor can occur in dimension four: use <Jordan blocks> of sizes at most two at <eigenvalue> $0$ and of size one at <eigenvalue> $1$, adding scalar blocks as needed. The complete list is
$$
\boxed{t,\quad t^2,\quad t-1,\quad t(t-1),\quad t^2(t-1)}.
$$
For example the first four occur for $0$, $J_2(0)\oplus0\oplus0$, $I$, and $\operatorname{diag}(0,0,1,1)$, respectively; the final one occurs for $J_2(0)\oplus0\oplus1$.
Failure to be <diagonalizable> requires a size-two <Jordan block> at zero. The condition $A^2\ne0$ requires at least one block at one, since every allowed zero block squares to zero. A size-two zero block and one scalar block at one leave only one dimension. It can carry zero or one. Thus, up to permutation of blocks, \b[the only possible <Jordan normal forms> are]
$$
\boxed{J_2(0)\oplus[0]\oplus[1],\qquad J_2(0)\oplus[1]\oplus[1]},\qquad J_2(0)=\begin{pmatrix}0&1\\0&0\end{pmatrix}.
$$
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