Back then, we used to have a plugin that allows users to show the badge image on their user cards like so
That plugin is deprecated because very, very few sites used it.
So, the dimensions don’t really matter much these days.
The only thing that matters is the aspect ratio - and it’s purely aesthetic.
Badge images/icons are rendered with a fixed 1:1 ratio in all the places they’re used. For icons, that’s not something you need to worry about.
If you want to upload an image. Stick with a rectangular image.
The largest badge image we display is 64x64, so double that for retina displays.
To avoid blurry images, I would recommend 128x128 as a minimum.
This should give you ideal results on all devices.