Is it conventional/understandable to denote the set of all $m\times n$ matrices over $F$ as $\left(F^n\right)^m$, as $m$-tuples of column vectors, just like $n$-tuples are used as $n$ dimensional column vectors? Or is $\left(F^m\right)^n$, row-wise, the convention?
I've seen $F^{m\times n}$, or even $F^{mn}$, which seem so ambiguous, as these just look like vectors of $m\cdot n$ dimensions.
Or are matrices represented an other way entirely? I'm just curious as to what the conventions of notation and representation are.
$F^{m \times n}$ is universally (among the mathematical community) understood to mean the set of matrices with $m$ rows and $n$ columns; the usage of the symbol $\times$ stands out and is rarely misinterpreted as simple multiplication. In fact, $\times$ is almost never used to denote multiplication except in the specific context of numerical arithmetic (as in $12 \times 11 = 132$).
None of the other notations you suggest would be understood in this way unless you provided further explanation. In fact, my first guess at the meaning of $(F^n)^m$ is that you are looking at block vectors consisting of $m$ blocks of length-$n$ vectors. Effectively, this amounts to seeing $(F^n)^m$ as $F^{mn}$ (vectors of length $n$), which is precisely the misinterpretation you're trying to avoid.
Another common notation that you may prefer (used for instance by Horn and Johnson in their texts) is $\mathcal M_{m,n}(F)$.