There is (currently) no easy way to mass select or mass delete your captures on Gyazo. If you would like to delete all your captures without much hassle, try this script. (Last tested on 1st of May 2020)
Copy paste the contents of gyazodel.min.js in a chrome developer console on https://gyazo.com/captures.
2 buttons will appear in the navbar.
Click the "Delete All" button to execute an automatic mass deleter for all visible items (this will take about 5 seconds to complete).
Click the "Select All" button to only mass select all visible items.
After deleting a couple of items, the Gyazo UI likes to get glitched. Reload the page and repeat the mentioned steps above.
Hey! Thanks for this. I've updated it and added another, much faster script that interacts directly with the Gyazo API, as you do when you click on the page. It's here https://gist.github.com/TCNOco/9fd372e229e32a7f442ebec45d8fa088
This is more focused towards those looking to delete most, if not all, of their images.
I have also updated the above to add the buttons, as the CSS selectors have changed a little.
Great script, regardless, even now.