I've made some bins for my CD collection (and larger ones
for DVDs) with drawer-like construction techniques. The
front and back are one-inch (nominal) lumber, but for sidewalls
and bottom, I've used tempered hardboard (like we used to call Masonite).

To hold the sidewalls to front and back, a rabbet and glue with some pin
nails does the trick, though assembling it isn't a neat process. Front
and back can take a dado to capture the bottom, but the sidewalls
are only 1/8" thick, so to hold the bottom, I made tab-in-slot features
with several 1" long slots in the sides, and with a dado blade cut into
the bottoms (as a clamped-up stack) to form the tabs.

The problem: hardboard isn't what my tools are intended for. A
1/8" diameter carbide router bit, and a Rotozip-style side-cutting drill,
make ragged slots (and it's not clear that the steel tool isn't overheating).
A table saw leaves a furry edge (OK, that cleans up, by hand, with
some sandpaper, but... it's a nuisance). I can apply a bit of shellac on
the furry edges, then iron them flat with the old Proctor-Silex appliance,
but one must be careful about the glue-surfaces.

Oddly, a brad-point drill makes a perfectly clean cut (to start the slot),
so there's ONE tool that isn't a mismatch to this material.

My two questions:
is there a way to form those tabs that doesn't turn
so much material into sawdust (probably not, but I've gotta ask),
and
HOW can I make a clean slot in what is essentially just heavy paper?
There's metal punch/die sets that could do it, but the supporting
press tooling isn't gonna fit in the basement.

I can imagine a nibbler tool to form the tabs, but bandsaw (I've tried it)
is slow compared to stacks fed to the dado blade in the table saw.