Every size a site actually needs — plus the manifest and the exact HTML to paste. No upload, no account.
Square works best. A 512×512 PNG with transparency is ideal; anything larger is fine.
Why these sizes and not thirty. Most generators emit dozens of files for
devices that stopped mattering years ago. What a site needs today is: a 32×32 for the
browser tab, a 180×180 for iOS home screens, and 192/512 for Android and the install
prompt. favicon.ico is included because some crawlers and older Windows
builds still request it at the site root and log a 404 when it is missing.
On transparency: iOS composites home-screen icons onto a solid tile, so a
transparent apple-touch icon shows up on whatever colour the OS chooses. That one is
always flattened onto your background colour — the others follow your setting.