You can view data across different browsers at Can I Use. However this information tends to confuse people as it looks as though IE is the only browser with support. Read on to understand the status of Grid Layout in browsers.
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.
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.
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.
My reference implementation is the one in Blink, unless stated otherwise on an example. I am typically using Chrome and Chrome Canary.
The Blink implementation has been carried out by Igalia, sponsored by Bloomberg. You can read about the implementation work in this post from Igalia.
At the present time you need to enable the Experimental Web Platform Features Flag in Chrome or Opera.
Download Chrome Canary if you want to get the very latest features that have been implemented. Chrome Canary does not need flags enabled.
There is no information as to when or if Apple will ship grid in Safari. A hopeful guess would be March, a somewhat more pessimistic one October, but this is just a guess based on release dates and the fact grid is not now behind a flag by default in Preview.
This implementation is also by Igalia but lags slightly behind the Chrome one. Note that Grid being present in Safari Technology Preview does not mean it will be in the next version of Safari. We can but hope.
Grid is currently being implemented in Gecko. You need to enable the layout.css.grid.enabled flag by going to about:config in regular Firefox. For the latest features download a Firefox Nightly or the Firefox Developer Edition. In Nightly and Developer Edition you do not need to enable the flag - Grid should just work!
Firefox Nightlies and Firefox Developer Edition include a grid highlighter in DevTools.
If you are using Firefox then you can use the experimental CSS Grid Inspector to help you visualize your grids. It is a tool in development - just like Grid Layout.