Vertical lines appearing in scanned documents

I've managed to (almost) solve the problem for myself. By removing a few screws, I could remove the left-hand side panel of the printer, giving me access to the underneath of the scanner mechanism. The scanner unit itself is almost a separate component, but I couldn't find how to access the screws holding it in place; however, there are three plastic clips holding the top half (the glass) to the bottom half (the assembly containing the scanner mechanics). By prising those up and jamming a couple of screwdriver bits underneath, I was able to separate the glass from the mechanism just enough to create a gap of maybe a couple of millimetres.

I blew some compressed air in (full disclosure: I actually used a kiddie's paddling pool inflation pump), which resulted in a pleasing amount of dust being blown back at me. A test of the flatbed scan was much better, but the ADF scan was still as noisy as ever - there still seemed to be some kind of dust or dirt on the underneath of the scanning glass used by the ADF. The gap in the housing wasn't big enough to get any kind of cleaning cloth in, and regular cotton buds (Q-Tips) were too short and too thick to reach the glass. Fortunately I have a few extra-long and narrow swabs handy ...

Running one of those on the underneath of the ADF glass didn't seem to dislodge anything obvious. However, looking at the results now, using the same black paper as on the original tests...

Flatbed scan

ADF scan

... it's almost perfect, with only the faintest of lines still visible. I've repeated the compressed-air-and-cotton-bud cleaning a couple of times, but this is as good as it's going to get I think - and on anything except the blackest of black areas the lines are now unnoticeable.

This obviously isn't a recommended fix for anyone except the most desperate - but it's quick and easy enough to try if "external" cleaning isn't making much of a difference. I suspect I may need to repeat this in the future: I do a fair amount of scanning, and there's clearly some way for a tiny amounts of dust to occasionally ingress into the scanning mechanism.