This scanner is almost 10 years old Canon CanoScan 8400F. It began giving showing some strange curly shapes to dark areas in images. Cleaning the glass didn't do anything. The pattern was on the other side of the glass. Dust or moisture seems to get there, apparently causing this.
This is a levels-enhanced scan of a black paper. The swirling shapes of the patinated underside are clearly visible. The top of the glass was not too clean either.

Scanner unplugged:

Lid removed. It comes out quite easily.

There are four screws to be removed.

The first two screws are in the lid's leg holes.

The other two screws are under the front cover. The cover snaps out by pressing from the edges where the clips are, and at the same time turning the front cover up or rown. Here the scanner is upside down:

The two screws on the top need to be unscrewed.

The top cover can now be detached. It is held in place by several plastic clips. A flathead screwdriver or a dining knife helps here. Twisting the frame too much risks shattering the glass.

Top removed, the glass can now be cleaned from both sides.

The dirt and patina was surprisingly persistent. Swiping with a moist cloth didn't really do anything. I first used Mellerud Adhesive and Glue Remover, then finished with lense cleaning liquid. Instead of cloth I used a piece of synthetic chamois. Turning the glass near a lamp helps seeing if it is clean.

Putting the scanner back together. A piece of blu tack helps in getting the screws to the lid's leg holes. These screws seemed to be non-magnetic.

Here's the same piece of black paper scanner again, levels enhanced to reveal any dirt and dust. Now the glass is very clean.

Before & after comparison:
