What are the standard sizes for desktop wallpaper?

To keep it simple and not bother with numerous legacy or the in-between sizes, I would go with:

1600x1024 (older)
1920x1080 (most common)
2560x1600 (largest)
* Desktop screen settings can do the stretch-to-fit if needed.

As far as Mobile, wouldn't your mobile-detect script just size the images accordingly & likewise on the device?

Don't forget the 5K display, mine is 5120x2880 at default. :)

In terms of aspect ratio would I be OK just considering 16:9 and 4:3?

> 5K
Egads.
Could I just make one massive 5120×2880 image for 16:9 and 5120×3840 for 4:3, and let all the other desktops downsize it, or are multiple versions within an aspect ratio a better idea?

I don't think it would need to be a full 5120x2880 because the pixels are smaller than on a typical Windows display. They are the same 326 pixels per inch as on the iPhone 4. Fonts and images are incredibly clear and crisp but it is not a literal translation because of pixel size (like device pixel ratios). I commonly view digital photos that are 3696x2448 and they are too big to view at full size on the 5K screen. At full size, those images have scrollbars.
Sticking to a 16:9 and 4:3 aspect ratio would be your best bet imo.

This is a complicated area. I thought it was going to be simpler than this.

StatCounter had some decent stats on screen resolutions [gs.statcounter.com...] So now I'm looking at a breakdown of the most popular resolutions roughly like this:

[16:9]
1366 x 768*
1920 x 1080
1024 x 768
1600 x 900

[16:10]
1280 x 800
1440 x 900

[4:3]
1024 x 768

[5:4]
1280 x 1024

I don't think I'll bother with 5:4 resolution. I'll probably just make large images in each of these ratios:
[4:3] 2048 x 1536
[16:10] 2560 x 1600
[16:9] 3840 x 2160