How to remove dust spots from a scanned photo
Dust Spot Finder in Paint by Zaweba finds dust on smooth areas of a scan, lists each spot with a tick box and heals the ones you tick.

Scanned prints and negatives nearly always pick up dust, and Paint by Zaweba has a tool built for it. Dust Spot Finder looks over the smooth parts of the picture, finds the specks, lists them with tick boxes and heals the spots you leave ticked. For the odd mark it misses, the healing tools or Filter > Noise > Dust and Scratches finish the job.
Dust shows worst in skies, walls and skin, where the eye expects an even tone. It is also tedious to fix one spot at a time, which is why a tool that finds them for you saves so much effort on a box of old family photos.
What you need
A scan at a decent resolution, Paint by Zaweba on a Windows PC, and a little patience for checking. Paint works offline, so private family pictures stay on your own computer. If you have a lot of prints to scan, it is worth cleaning the scanner glass and the prints first. Less dust in means less work later.
Step by step
Open the scan and run Dust Spot Finder. It searches the smooth areas, which is where dust stands out and where it can be found reliably. It can show the spots it found on the picture, so you can see what it is about to change.
Then go through the list. Every spot has a tick box. Untick anything that isn't really dust, such as a catchlight in an eye, a distant bird or a button on a coat. When you are happy, heal the ticked spots and they are filled in from their surroundings.
Afterwards, zoom to 100 percent and pan slowly across the picture. The finder deliberately works on smooth areas, so specks over busy texture, such as foliage or patterned fabric, need a manual touch.
Fixing what is left
For single spots, the Spot Healing Brush is quickest: click on the speck and it blends texture from around it. For marks on a detailed area, the Healing Brush lets you choose where to sample from, so the texture matches. Long thin marks are a different problem, and the Scratch Remover handles them: drag along the scratch and the strip is rebuilt from both sides.
If a scan is peppered with fine dust everywhere, Filter > Noise > Dust and Scratches treats the whole picture or a selection at once. It works by smoothing out small details, so use it with care. A good habit is to run it on a copy of the layer and mask it in only where it is needed, such as across a plain sky.
Getting a good result
Work on a duplicate layer so the original scan stays untouched beneath. Check the healed spots against the original by hiding and showing the top layer, because a heal that looks fine at a glance can leave a soft patch that shows in print.
Don't chase perfection on old photos either. Film grain and a little age are part of their character, and an over-cleaned print can look plastic. The aim is to remove what distracts, not to make the photo look new.
Doing a whole box of prints
If you scanned several prints on the flatbed at once, File > Automate > Split and Straighten Photos opens each print upright as its own document, ready for cleaning. Paint by Zaweba also records actions, so any repeated steps can be replayed on the next scan instead of being set up again. Dust removal itself is still best checked by eye, one picture at a time, but the work around it can be made quick.