I've been starting to think about the problem of using the sketchbox dice tool for an improvised underworld, similar to using it for improvised towns and villages. Most of what I've described so far deals with using the dice tool as an aid to prep, but if we want to truly leave most of the underworld vague or unmapped, we need a different approach.
Step One: Write down one to three general statements about the underworld. These should just be short phrases, like "unhallowed necropolis of the demon-prince" or "fallen city from centuries ago". One of those statements should suggest the reason why the underworld fell, such as "flooded by slime" or "buried by earthquake". The exceptions would be if it's merely old and forgotten (abandoned because of climate change or economic factors,) or if the underworld was never above ground and was built by its current occupants. Another phrase or statement could suggest current distinguishing features, like "tainted by visible, glowing evil".
Also pick a starting size (equivalent to hamlet, village, town, city, metropolis.) This, in combination with the statements above, will give you a general grasp of what's going on down below, enough that you could start a vertical level cross-section.
Step Two: Write down a couple nearby/connected underworld areas. A one-phrase summary for each of these will do for now. These can be on the same level, connected by tunnels, or below the top level, connected by pits, chutes, ramps, and stairs. If you are doing a cross-section, add these where desired.
Step Three: Write down one to three general statements about each quarter of the starting underworld. One of these statements should say what is unique about the appearance of that quarter, such as the typical size/shape of a commoner's home and how roads are constructed.If there is a Focal/Special quarter, it will have an original purpose. If the current occupants are different, you may want to convert a different area to cover a new main purpose.
Step Four: Roll for or select the state of decay or ruin for each quarter and each connecting tunnel, including tunnels leading to other levels or nearby underworlds. For any quarter connected to the surface, use a base 2d6 roll, +2 if you intend the level to have intelligent occupants, -2 if you want to start with a more abandoned feel. Then, roll for any tunnel connected to one of these starting areas; use the unmodified roll if the adjacent area is Average condition, add +1 to the roll for every adjacent area in Good condition (+3 per Very Good,) subtract -1 from the roll for every adjacent Bad area (-3 for Very Bad.) Continue to roll for areas these tunnels connect to; if a tunnel is blocked/collapsed, only roll for the immediately-adjacent, isolated area, then stop; you can leave areas connected to an isolated area until later.
(Step Four-Point-Five: If any tunnels are sealed, decide how they are sealed. If the tunnel leads to another sublevel (adjoining underworld,) it's probably leagues long, so consider using the Lake/River or similar obstructions. It's important to do this when necessary before Step Five.)
Step Five: Build your wandering monster lists for each quarter. The decay status of the quarter tells you what kind of monsters might work best, as will the method of obstruction for some tunnels. Areas that are close to each other will probably have overlapping monster types. More distant areas will have little or no influence.
These steps give you the basic concept of your underworld. I'll continue in Part Two.