I have had the problem of PDFs not opening since a few weeks after upgrading to Catalina, and I can report that garycmartin's second suggestion works as a temporary workaround. Note: my PDFs have always been set to open in Preview, which is the default (not Affinity Photo, Adobe Acrobat or anything else), so that is not the problem.
Specifically, if I double-click any PDF file (or open Preview and then File...Open...), then I would expect a new window to open up with the PDF file showing, but instead I get nothing. Preview opens, as is clear from the menu bar, but there is no window, and the PDF is not rendered. Interestingly, the file does show in Finder under "Quick Look," just not under Preview.
To workaround this problem, open Terminal (it is an app in the Applications/Utilities folder), type the following command line, and hit return:
/System/Library/Frameworks/CoreServices.framework/Frameworks/LaunchServices.framework/Support/lsregister -kill -r -domain local -domain system -domain user
Then restart your computer. This will fix Preview for a month or two, after which the problem will recur. I have had to apply the fix about half a dozen times since Catalina.
I hope that someone at Apple is reading this forum and will find this workaround helpful to debug the problem, which has persisted for an inexcusable duration!