How often does Google update its:
1. Backlinks?
2. PR?
3. Search results?

1. Backlinks - usually just before a PR update
2. PR - On average 4 times year
3. search results - pretty much continuously.

Like stOx said. They update search results continuously. PR and backlinks is about every 3 months, but in truth it's whenever Google decides to update. There's no schedule though based on the history of updates another one should be coming in the near future.

I guess the latest 3-in-one update took place several days ago.

They updated their PR and backlinks on the 13th of this month and they totally missed mine!