How many miles under the ground do the Underdark tunnels and caves typically go? One? Five? A hundred?
At least 26 miles deep, with rumoured unexplored caverns 40 miles deep, though most of the known Underdark is within 10 miles of the surface.
According to the D&D 3e sourcebook Underdark, p.120, the lowest level of the Underdark starts at a depth of ten miles and continues down from there:
No place on Toril is as strange and dangerous as the Lowerdark. This level of the Underdark extends from 10 miles below the surface to unfathomable depths and features a degree of strangeness that would drive some surface dwellers insane.
At this depth, it is described that tunnels and passageways become rarer. Since this is the bottom of the Underdark, the typical depth of Underdark locations is between 1 and 10 miles beneath the surface. For example, the city of Menzoberranzan is only 2 miles below the surface, and is considered part of the Upperdark.
Portals to the plane of Shadow (what would in 5e be called the Shadowfell) are stated to exist at depths of 15 miles or below. A sea flows 20 miles below the surface (p.123):
The Glimmersea lies 20 miles below the floor of the Sea of Fallen Stars.
Even deeper is the Genasi city of Earth's End (p.147), although it cannot be reached without teleportation and has no direct tunnels:
Situated nearly 22 miles beneath the Nath in Halruaa, Earth's End is unapologetically designed for its extraplanar visitors.
Oryndoll, City of Loretakers, is just over 26 miles deep (p.168).
There's also an abyss beneath Anauroch, named Lorosfyr, the Maddening Dark, rumoured to be at least 40 miles deep in places (p.160), though it it unexplored.