Current Browser/Rendering Engine Information
You can view data across different browsers at Can I Use.
If you find a bug in a browser implementation then the best thing to do is to search the logged bugs for that browser as this will include known bugs and issues still being worked on. If you have found something new then you can find information about logging bugs with browsers here.
Gecko Rendering Engine (Firefox)
Firefox were first to land grid support, on the 7th March 2017 in Firefox 52.
A Grid Layout tool for Firefox
Firefox includes a grid inspector in DevTools.
Blink Rendering Engine (Chrome, Opera)
Chrome 57 has now shipped with Grid Layout included. The listed date was March 14th 2017, however 57 was updating in browsers at the end of the previous week.
The Blink implementation has been carried out by Igalia, sponsored by Bloomberg. You can read about the implementation work in this post from Igalia.
Safari 10.1 and iOS Safari support CSS Grid as of March 27, 2017.
On launch Edge shipped with the IE11 Grid implementation. Therefore the current implementation in Edge is against the original and not the new spec.
On Monday 3rd April, 2017 - on stage at An Event Apart - Microsoft announced that work has started to update the Edge Grid implementation to the new spec. Thank you to everyone who voted for the feature, it demonstrated that this was something we wanted.
Internet Explorer 10 and 11
The original Grid implementation was part of IE10, also in IE11. This implementation is now quite different to the current spec and other implementations. Internet Explorer 11 will never be updated to the new specification. There is some possibility in using the old spec to provide some grid layout for these browsers however.