The Mines Abad has created a useful plugin that removes the ‘category’ label from WordPress category URLs.
You can get it from the WP plugins repository:
After installing it, I can see the category URLs changing from:
I tested it locally and it works fine for me. The author is looking for more feedback and testing in different environments. You can help his efforts by trying it out and reporting issues on the plugin page.
After activating or deactivating, you might want to go to Settings->Permalinks and resave your site’s permalink structure. This flushes all rewrite rules.