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.
Firefox were first to land grid support, on the 7th March 2017 in Firefox 52.
Firefox includes a grid inspector in DevTools.
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.
On the 24th January 2017, Apple announced the features that would be part of the Safari 10.1 Spring update now in beta. This includes Grid!
On launch Edge shipped with the IE11 Grid implementation. Therefore the current implementation in Edge is against the original and not the new spec. In September 2015 Microsoft indicated that updating Grid to the current spec is on the backlog with a priority of High. Do add your votes to encourage implementation.
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.