How to fix the lines in interlaced video
Comb-like lines on moving edges mean interlaced video. Tick deinterlace in Cut by Zaweba's Filters tab and save, and the lines are gone.

Those thin horizontal lines along moving edges are interlacing, and Cut by Zaweba removes them. Open the video, tick deinterlace in the Filters tab and save it through the Convert tab. Cut uses FFmpeg's bwdif deinterlacer, which rebuilds full frames from the two halves the video was recorded in.
Cut by Zaweba is free software under the EULA. It runs offline on Windows, so family footage stays on your own PC.
What interlacing is
Older television systems didn't send whole pictures. They sent every other line in one pass and the remaining lines in the next, a fraction of a second later. On a tube television the two halves blended together and nobody noticed. Many camcorders, especially those recording to tape or to MTS files, still recorded this way.
On a modern screen, both halves are shown at once. Where nothing moves, they match and the picture looks fine. Where something moves, the two halves were captured at slightly different moments, so the edge breaks into a comb of alternating lines. Pans, running children and passing cars show it worst.
Checking whether a video is interlaced
The quickest test is your own eyes. Step frame by frame through a part with movement using the left and right arrows; if you see combing on moving edges but not on still ones, the video is interlaced. Cut's preview frames are exact in every container, so what you see is what's in the file.
The Details tab is useful too. It lists the video codec, resolution, frame rate and pixel format, among other things, which tells you what sort of camera or recorder the clip came from.
Removing the comb lines
Open the file in Cut. It reads MTS, M2TS, TS, MPG, AVI and the other common formats, so camcorder and recorder files open as they are.
Trim first if you want to. Mark in with I and out with O, or mark an unwanted stretch and press Delete. Old recordings often begin with blank tape or end with a minute of the floor.
Go to the Filters tab and tick deinterlace. To see the difference before saving, turn on View, then Show the filters in the preview, and step through a moving section again. The comb should be gone.
Then open the Convert tab and choose a preset. Best quality keeps the most detail. For YouTube 1080p, For WhatsApp, Small file and For email suit particular destinations. If the video is going back to an old television or DVD player, there's a preset for that too, saving MPEG-4 ASP in an AVI file tagged XVID, up to 720 by 576, with MP3 sound. The estimated size and encoder are shown before you start.
Mistakes that make it worse
Deinterlace once, early. Don't convert an interlaced file with another tool and then try to fix it, because resizing interlaced video first smears the two halves together and makes them much harder to separate cleanly.
Pair it with denoise for old tapes. Footage that's interlaced is often grainy too. Denoise, light or strong, is in the same tab. Start with light.
Don't deinterlace video that isn't interlaced. Phone clips and most modern recordings are already whole frames, and the filter has nothing to fix there.
A whole box of tapes
If you've captured a stack of old tapes to files, set deinterlace and any other filters you want, then add all the files at once from the Batch tab. The queue works through them in the background with progress bars, two jobs at a time, or one in low-spec mode on a modest PC. It's saved on every change, so a restart halfway through a long evening doesn't lose your place.
Interlacing is one of those problems that looks complicated and turns out to be a single tick box in Cut by Zaweba.