It is true that the (vast?) majority of math-script and math-calligraphic font shapes provide only uppercase letters. For some math font packages that provide these font shapes for lowercase letters as well, please see the pages 8 thru 10 in the user guide of the mathalpha package.
Do also note that the terminology of what's a "calligraphic" or a "script" font shape isn't fully standardized. One working definition is that "script" letters are more heavily sloped than "calligraphic" letters are; however, do be aware that this is just one, and certainly not universally accepted, possible definition.
After perusing the user guide of the mathalpha
package in the manner suggested above, I came up with the following choice: esstix
for \mathcal
and \boondox
for \mathscr
. Again, let me emphasize that this is just one particular possible choice. Others may come up with other choices.

\documentclass{article}
\usepackage[scr=boondox, % heavily sloped
cal=esstix] % slightly sloped
{mathalpha}
\begin{document}
$\mathscr{abcABC} \quad \mathcal{abcABC}$
\end{document}