There are SO many different ways that people like to run leagues and tournaments that there's simply no "best" software tool for the job. But after playing with several, these are the ones that I've used personally and enjoy. Click the logos to visit the various sites.
This tool has been growing steadily for at least a couple of years, and it has grown into one spectacular app for running pickleball events! We use Pickleheads for all of our leagues and tournaments.
This is a great tool for handling ladder leagues, which they call "Shootouts". GPC used Den for years and it worked well for us, but we moved to Pickleheads because I found their interface to be better, and the cost is lower. Den includes club memer management tools whilc Pickleheads doesn't, so you'll need to weight the value of that for your organization.
This is a HIGHLY POLISHED site with tons of pickleball features, and it's FREE. Wow! You'll have to advertise elsewhere, though, as you probably won't reach enough people through GPN alone to fill a tournament...unless you do small ones like me, then it's fine. :)
You can run leagues (with their own ratings calculator), but honestly the data entry is cumbersome enough that I much prefer Pickleball Den.
You can run tournaments, and I've been very happy with the round robin tournaments I've run with this. I've not tried elimination brackets, though I'm sure they work smoothly, too.
If you want a very traditional league (fixed partners, playing for a fixed number of weeks) then this is SPECTACULAR! Seriously, the user interface is second to none. I LOVED using it. They aren't pickleball people, though, so the concept of rotating partners is lost on them, but for any traditional league, they're hard to beat! They offer tournaments, too, but I've not tried this, and honestly I feel like a pickleball-specific tournament tool will do a better job (printing score sheets, etc.).