I've put a Pfaff 130 in a Singer cabinet without a problem.
Not sure what size a 360 is. And sometimes it depends on the cabinet - some have the lift spring, and some have a different front for the 301 and later machines.
I'm not sure what you mean by "generic Singer cabinet" Unless it's one that was designed for a Singer Clone type machine, so it's not a Singer cabinet, but most Singers will fit in it, along with the Japanese made clones (and quite a few later Japanse models too.