MS-DOS had TSRs (terminate and stay resident) programs, but those aren't exactly the same thing. I do recall some basic multitasking utilizing them in some way, though. My memory is too fuzzy.

MS-DOS 4.0 had a multitasking release, but it was not popular and only available through retail channels. IBM didn't want it so there were OEM MS-DOS releases of 4.0 that didn't have multitasking, and that's what you got when you bought an IBM PC. This meant it didn't get a lot of support and was discontinued after MS-DOS 4.1. If it sounds like a mess, it's because this was all when IBM and Microsoft were not very happy with each other, which is kind of why everyone mostly jumped from MS-DOS 3.3 (IBM and MS mostly cooperative) to MS-DOS 5.0 (MS independent).

DOSSHELL supported task switching, but only one process could execute at a time, and all tasks had to fit in memory.