The current version of Firebug (2.0.13 + FF43) seems to debug typescript just fine. At least it has worked for me so far.
On the web page, you add a βcompiledβ .js with a link to the source map. I.e.
<script src="register.js"></script>
And Firebug will show you the register.ts file instead in the script list.
Make sure your tsconfig.json has source map generation:
"compilerOptions": { ... "sourceMap": true },
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