From my research for stockfish.wasm:

Chromium based (desktop only):

• Since 79: Full support
• Since 74: Support, but without growable shared memory
• 70 to 73: Needs flag chrome://flags/#enable-webassembly-threads or Origin Trial

Firefox:

• Since 79: Full support
• 72 to 78: Structured cloning can no longer be enabled with flags, except on nightlies
• 71: Requires javascript.options.shared_memory and dom.postMessage.sharedArrayBuffer.withCOOP_COEP to be enabled in about:flags
• 68 to 70: Requires javascript.options.shared_memory to be enabled in about:flags

Other browsers: No support.

Updated 2020-11-05

Along with SharedArrayBuffer being re-enabled, WebAssembly Threads are also available since Firefox 79.

Threads in FF 79 (and Chrome Android / M91 desktop) will require the COOP/COEP headers as a mitigation for Spectre-style side-channel attacks: https://web.dev/coop-coep/

NodeJS supports threads as of v12.0.0 but requires the v8 flag --experimental-wasm-threads.

Seemingly available in Safari 14.1 https://webkit.org/blog/11648/new-webkit-features-in-safari-14-1/#:~:text=WebAssembly-,threading,-WebAssembly:

In this release, WebKit added WASM threading to support CPU-intensive applications that need to work across CPU threads to preserve application performance.

Not sure about TP, didn't find it in the release notes…

Webassembly Threads still not supported in iOS browsers, sadly. Does anybody know when this might be coming?

It's been supported since iOS Safari 15.2 with COOP/COEP enabled IIUC.