It's not rare, it's not common. It's one of a number of synonyms for "to think." It suggest that you are thinking for a long time, in a relaxed and unhurried way, trying to make a complete analysis. In addition to "to ponder," we have "to muse," "to mull over," "to puzzle over," and "to think through" (as opposed to simply "to think.") My personal test for whether something is fairly common in everyday English--as opposed to literary, formal, or academic English--is to try, not a Google search, but a Google _News_ search. Google's news sources are mostly professionally-written "good English," but at a level that ordinary people are supposed to understand. It is educated, adult English but not overly formal or flowery. I found many examples: "Crews Stabilizing Highway 1 in Big Sur As Engineers Ponder Long-Term Solution" This clearly suggests that the crews are working quickly, while meanwhile the engineers are taking their time doing a thorough, completely analysis. "Are we in the golden age of gaming adaptations? We ponder the question on the latest Gaming Gossip." "Jeff Weltman, Orlando Magic shoring up the margins as they ponder their future" The word is expected to be familiar to sports fans. "Dollar takes a breather as investors ponder US rates outlook"

Oh, we use it in daily speech fairly often. At least in SA and my circles in the UK.

I’d use it in the infinitive or in a command: Here’s something to ponder- … Let’s ponder that and discuss it tomorrow..

"Ponder" is formal and I feel like it has an association with intellectual topics like philosophy. I usually use "think" and "wonder" in normal speech

"Ponder" is a bit formal in tone, so I use "think about", "wonder if" or "wonder whether" much more often in everyday speech. If you want another choice that's even more formal in tone, you could also use "contemplate."