For an $m\times n$ matrix $A$ of rank $r \ll \min(m, n)$. $A$ has $r$ positive singular values: $$ 0\le \sigma_r \le \ldots \le \sigma_2 \le \sigma_1. $$ Denote as $\sigma\in \R^r$ the singular value vector. Its norms would satisfy:
Spectral norm $$ \Vert A\Vert _2 = \sigma_1=\Vert \sigma\Vert _{\infty}. $$
Frobenius norm
$$ \Vert A \Vert_F = \sqrt{\sum_{i=1}^r \sigma_i^r} = \Vert \sigma\Vert_2 $$
Nuclear norm $$ \Vert A\Vert_* = \sum_{i =1 }^r \sigma_i = \Vert \sigma\Vert_1. $$
$\Vert A\Vert_F $’s order will be $O(\sqrt{r} \sigma_{\max})$, and $\Vert A\Vert_*$ will be of $O(r \sigma_{\max})$. Actually
$$ \Vert A\Vert_2^2\le \Vert A\Vert_F^2 \le r\Vert A\Vert_2^2. $$ Further assuming $|A_{ij}|\le \bar a$ (uniformly bounded) we do have $$ \Vert A\Vert_F\le \sqrt{mn}\bar a. $$