Perhaps a more probabilistic example:
For each $n$, let $Z_n$ be uniformly distributed on the set $\{1, 2, \dots, n\}$. For $1 \le k \le n$, let $X_{n,k} = \begin{cases} n^2, & \text{if $Z_n = k$}\\ 0, & \text{otherwise}.\end{cases}$ Then consider the sequence $\{X_{1,1}, X_{2,1}, X_{2,2}, \dots, X_{n,1}, \dots, X_{n,n}, \dots\}$ which steps through the "rows" of this triangular array. It converges to 0 in probability since for any \epsilon < 1, we have $P(|X_{n,k}| > \epsilon) = P(Z_n = k) = 1/n$ which goes to $0$ as $n \to \infty$. It does not converge almost surely, since the sequence will always have infinitely many values greater than 1 (one in each row) but all the rest are zero. Finally, we have $E|X_{n,k}|^p = \frac{1}{n} n^{2p} = n^{2p-1} \to \infty$ for all $p \ge 1$, so the sequence does not converge in $L^p$ either.