This paper by Darmon and Merel, which proves the general case $x^n + y^n = z^3$ has no primitive solutions, cites Dickson, "History of the theory of numbers", page 630. For a good resource on the state of the art on the generalized Fermat problem (well, as of 2005), see Poonen, Schaefer, Stoll, "Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$". This is where I found the Darmon-Merel reference. If someone knows a good reference with a nice summary of what's known as of a more recent date, please let me know.
As a side note, my impression is that there are many professional mathematicians do not prefer the name Beal's conjecture for this; I have occasionally seen a parallel problem called the "Generalized Fermat conjecture": that $x^n + y^m = z^\ell$ has finitely many primtive solutions $(x,y,z,n,m,\ell)$, where $1/n + 1/m + 1/\ell < 1$ (other than the ones that obviously work for an infinite family of $(n,m,\ell)$; e.g. $(2,\pm 3, 1, 3,2,\ell)$. You will not find the name Beal's conjecture in reputable published papers on the problem, for instance.