There is no Test suite column when browsing Test cases, but if you open up an individual test case, you will see "Related Test suites" on the right side panel.
To add to my earlier comment, if a column is available for display in browse artifacts view, you will see it under "Change column display settings" at the top right as shown below.
Dashboards and work items are no longer publicly available, so some links may be invalid. We now provide similar information through other means.
Learn more here.
