Yes you could under very controlled circumstances, but in practice just use SSE2 and/or MMX instead.
Related: Is it valid to write below ESP? discusses things in Windows that can asynchronously use the stack pointer in 32-bit code. With an invalid stack pointer, those things would crash instead of stepping on space below it. (Or if pointing to writeable memory, use that as stack space.)
In GNU/Linux, signal handlers can asynchronously use the user-space stack pointer, but you can use sigaltstack / SA_ONSTACK to use an alternate stack for them.
Also note that x86-64 guarantees SSE2. You usually only need to consider using RSP as a 16th general-purpose register if you've already used all of xmm0..15 (SSE) and mm0..7 (MMX).
Using ESP as an 8th general-purpose register in 32-bit DSP code for CPUs without MMX made sense sometimes; that's why discussion about it can be found in the context of virtualdub filters.
It generally doesn't make sense in 64-bit code because you always have 16x 128-bit SIMD registers (and SIMD instructions to use on them), as well as more than twice as many non-stack-pointer integer registers. And 8x 64-bit mmx registers, or 8x 80-bit x87 registers, however you want to use them. In most calling conventions, most of these registers are call-clobbered, but you couldn't make function calls with RSP not pointing into the stack anyway.