Hm. I'll start with something I should have added: what I'm currently working with. Also preferred budget.
Currently I'm using a Fluke 199C Scopemeter as both my scope and my 'bench' meter (I have other meters, but those are field units that live in my toolbag: Fluke 289 and 377). Instek GPD-3303S power supply. No microscope yet, just an el-cheapo armature light/magnifying lens.
"Do you want recommendations and actual make and model of various equipment or just a list of useful devices?": I'll take both please? With a weighted desire for specific make/model recommendations.
What I work on: Misc consumer electronics repair (for personal hobby / getting stuff cheap), commercial low voltage stuff (access control mostly - for work), battery pack repair (Milwaukee M18s - that's the tool ecosystem I use), repair of cheaply acquired 'for parts' stuff on eBay that will contribute to the lab (currently fixing a BK Precision 1786B programmable power supply). Interested in getting into/learning Raspberry Pi in the future since I see them integrated in so many things.
Budget: looking to spend under $1K ea for the scope and microscope (a trinocular or basically any microscope solution that gives me video output, preferably HDMI with a nice camera instead of USB with a crappy camera), under $500 ea for the DMM and power supply. Minimum 5.5 digit DMM, don't think I'll ever need more than 6.5 digit (and above that is absolutely out of range for what I'd want to spend anyway).
But with all of it, I want to keep an eye to having everything be compatible with a single software solution (if that is possible) to integrate all the data logging, realtime viewing, and control functions onto a computer. I would want to avoid, say, Scope A (that only works with software A) and DMM B (that absolutely won't work with software A).