  We show that the subcritical $d$-dimensional nonlinear Schr\"odinger equation
$i \psi_t + \Delta \psi + |\psi|^{2 \sigma} \psi = 0$, where $1<\sigma d<2$,
admits smooth solutions that become singular in~$L^p$ for $p^*<p \le \infty$,
where $p^*:=\frac{\sigma d}{\sigma d -1}$. Since $\lim_{\sigma d \to 2-} p^* =
2$, these solutions can collapse at any $2<p \le \infty$, and in particular for
$p = 2 \sigma+2$.
