Singular

Already ring r=0,(x,y,z), (c,dp); or ring r=0,(x,y,z),(C,dp); are big improvements for me. So maybe you can tell me where to look to get "signature-based" or Schreyer orderings in Singular, as I can't seem to find them mentioned in the online manual.

You can have POT or TOP for free resolutions - depending on the ordering of the ring
(therefore a ring in singular has always also a module ordering, even if only ideals are defined in it).
There are also different algorithm to compute free resolution, some of them use internally a diffirent ordering, for example fres (Schreyer ordering or 'signature based'), lres (dp,c or TOP), etc.

Maybe someone knows why TOP seems to have been chosen as the only module ordering to use in free resolutions of ideals and modules. I would have though the default that makes most sense as a mathematical default would have been POT, but I would rather use a signature-based ordering in the ideal case, which means each new map in the free resolution has a slightly different ordering based on the names (signatures) of the module generators. (As an aside, I am not a fan of minimizing here, just as I am not a fan of minimizing as done in normal.lib, since it is not really mathematically driven, only minimization-driven.)