I usually remember what Frobenius norm is. That is, the sum of square of all elements in a matrix. And I sometimes remember what nuclear norm is. That is, the sum of singular values of a matrix. But I forgot what trace norm is when I came across it again today.
Actually all of them are special cases of Schatten norm, the pth root of the sum of pth power of all singular values. When p=2, Schatten norm ends up equal to the Frobenius norm as
$latex \sum_{i,j} A_{i,j}^2=\sum_{i,j} A_{i,j} A^\top_{j,i}=\sum_i (AA^\top)_{i,i}=tr(AA^\top)=tr(U \Sigma V^\top V \Sigma U^\top)=tr(\Sigma^2)$
if $latex A=U \Sigma V^\top$ by svd.
And it turns out trace norm is just another name of the nuclear norm (Schatten norm when p=1) and is equal to $latex tr(\sqrt{A A^\top})$ as
$latex tr(\sqrt{A A^\top})= tr(\sqrt{U \Sigma^2 U^\top})=tr(U \Sigma U^\top)=tr(\Sigma)$.
Another norm I tend to get confused with is the spectral norm. It is a special case of operator norm or induced norm, where
$latex \|A\|_p=\sup \{\|Ax\|:x\in K^{n}{\text{ with }}\|x\|=1 \} =\sup \left\{ \frac {\|Ax\|}{\|x\|} :x\in K^{n}{\text{ with }}x\neq 0\right\}.$
Spectral norm is just the induced norm with p=2. In that case, it just ends up to be the largest singular value.
N.B. by convention, the singular values should always be positive. We can achieve that by adjusting $latex U $ and $latex V$ accordingly.