You can see the currently open tickets across all categories by creating a menu item of the Latest Open type. See https://www.akeebabackup.com/documentation/akeeba-ticket-system/component-frontend.html Besides that there's no reporting.
Think about the structure. You have a table with the tickets #__ats_tickets and a table with the posts #__ats_posts, with an one to many relation between them (one ticket -> many posts). The posts have a one to one relation to Joomla's users table #__users. Any kind of reporting would require to do joins between these three tables. That's a very, very slow process. If you have 1,000 tickets with 10,000 posts and 100 users the join creates a matrix that's 1,000 x 10,000 x 100 = 1,000,000,000 rows in the worst case. Usually MySQL will use the one-to-one relation to simplify the matrix so it'll only result in 1,000 x 10,000 = 10,000,000 rows. Now if you try running aggregate functions and grouping on them you put quite some strain to the database. And that's on a relatively "light" ticket system. If you try applying that to a mature ticket system with a decade's worth of data - like ours - you're basically killing your database server. So we chose not to implement complicated reporting since any attempt to use would bring your server down. Trust me on that. Been there.
What you can do, relatively fast, is queries on the #__ats_tickets table since it already contains most of the data you will need to report. Interactive search to that table is performed from the backend Tickets page. Here's a cool trick. Go that that page, apply the filters you want and then click on the Tickets link again. Tack &format=csv&limit=0 to the URL in the address bar to get a CSV file with all the records. Please remember to access the page again with &limit=10 to prevent Joomla! from remembering the "show all records" limit which might screw things up. The CSV file can be imported to Excel for formatting. Alternatively you can use format=json to get the output in JSON.
If, however, you want to export a single ticket with all its replies just view that ticket in the frontend and click on its Print Friendly View icon, right below the ticket number. If you only know the ticket number there's another cool trick, as long as you also know the link to your site's ticket system (the All Categories menu item to ATS). Let's say that your ticket system lies in http://www.example.com/support.html and you want to see ticket 1234. Type the URL http://www.example.com/support/1234 and ATS will redirect you to the correct page for the ticket. You can use that trick in your report to link to the ticket - basically, a parametric link using information you already have in the tickets report you exported to Excel. Links would appear in a PDF export.
Nicholas K. Dionysopoulos
Lead Developer and Director
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Please keep in mind my timezone and cultural differences when reading my replies. Thank you!