My D&D multiverse is basically the Planescape version (when I don't want to use D&D setting IP, I play something other than D&D).
In general I treat personal appearances as rare, but signs, omens, or manifestations of divine power as less so. For instance, in Ghosts of Saltmarsh a sidebar mentions Procan messing with people who loot shipwrecks. If this had come up I would have rolled a random chance of him doing something, and if so had it be something like unusually bad luck with weather or monsters while sailing for a month or so.
If you head to the Outer Planes you could theoretically gain a personal audience with a deity, and seeing an avatar would still be rare, but less so.
In general, the Planescape/D&D assumption seems to be that while deities might communicate with individual followers, they mostly let their religions just act on their own rather than overtly directing them like a workforce or army. That seems rather odd, but I think diverging from it turns the whole setting into a strategy game played by the gods, which is probably why they don't do that by the default assumptions, and why I'll probably stick with it. I suppose a reasonable justification might be that they gave their religion some teachings originally, and as long as the religion overall is more or less doing that sort of thing, it is advancing their portfolio and doesn't need micromanaging.
In my current campaign, the tiefling (non-standard) PC's backstory is that they are a distant descendant of Beshaba, who is also his warlock patron. She communicates with him in dreams (which we play through on-screen) and sent an avatar to physically interact with him in his backstory.
I'm also considering revamping the Piety rules along the lines of the Theros book, because the tiefling has piety and I've been using a modified form of the DMG version.
In some cases, deities might manifest an active interest working against, in favor of, or parralel to the PCs, by the dirt's if means mentioned, but whether or not that happens is highly campaign dependant.
So basically, some general guidelines about how things work in the setting, with individual campaign divine activity being highly variable based in the individual campaign.