Seems like your ground is crap... which is probably a probing problem still?
Routes under say 10" long, with common HC CMOS / TTL level signals, unlikely to have problems. A board that size, with stitched ground, should look a lot better than that, and any squigglies should be much shorter (~1ns). None of your scope shots are nearly fast enough to be able to view that sort of stuff; you're seeing crosstalk from something, or maybe some really nasty supply bounce (not as well stitched as you thought, or bypassed?).
Signals on board are generally very clean. It's hard to get much trash on anything routed within a PCB, unless it touches a connector, or runs at high speed (sub-ns edges -- needless to say, you need a scope with sub-ns risetime to pair with it).
The point where signal quality demands multilayer construction is going to be high speed (LVCMOS and faster, ECL, LVDS, etc.) and either high density (no room to stitch ground around the controlled-impedance traces, which have to be rather wide on a two-layer board), large area (e.g., backplanes, computer boards, etc.?), or both. In such projects, routing demands almost always take priority -- an FPGA isn't much use to you if you only have two layers to fan out its pins on, for instance.
Tim