Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra?
Weyl algebra computations are implemented in Macaulay 2 (package Dmodules). Here is a manual.
I use GAP (http://www.gapsystem.org/) with the GBNP package (http://www.win.tue.nl/~amc/pub/grobner/chap0.html). You define Weyl algebra as a quotient of free associative algebra by the ideal $<x_1 x_2  x_2 x_1  1>$. Easy to generalize to arbitrary dimension and can handle Clifford algebras (of arbitrary signatures) in the same setting.

$\begingroup$ Have you used singular? I have been trying to install and I am struggling. $\endgroup$ – No1729 Sep 2 '15 at 11:51

$\begingroup$ Struggling in what way? Where are you installing it? $\endgroup$ – Igor Rivin Sep 2 '15 at 11:53

$\begingroup$ I have followed the GUIDED CYGWIN INSTALL instructions on the site you linked. The mirror they suggest doesn't seem to up anymore. I have tried other mirrors but in the end when I try to run Singular out of the Cygwin window I just get error messages. $\endgroup$ – No1729 Sep 2 '15 at 11:56

5$\begingroup$ If you install Sage, you get a copy of Singular included with it, and the Sage install process is quite carefully tested on a wide range of platforms, so it's possible that this may be easier than installing Singular alone. $\endgroup$ – David Loeffler Sep 30 '15 at 21:15

$\begingroup$ you can run singular on cloud.sagemath.com $\endgroup$ – Dima Pasechnik Oct 1 '15 at 4:13