I've just scrolled through the Vatican Museum/Sistine Chapel ticket offerings on the official website. I purchased what must have been one of the last few tickets available for February 13 about an hour ago. At this point the only dates with basic entry tickets available between now and the end of March are February 8-10, March 15, March 17 and March 26. Many other dates do have tickets available for the Vatican-operated early-access tours.
Given the extremely limited availability of tickets, I'm surprised to see Feb 8-10 still for sale. I suspect those are tickets turned back by tour groups. Of those 3 dates, only Feb 10 is labeled "Low Availability".
I see there are some tickets available at 2:30 PM on April 8, the day before Easter. That's surprising, too, given that the period April 1-7 is sold out for the lowest-cost entry tickets. Perhaps it's a timing thing; I believe those tickets went on sale recently.
It looks like we need to keep warning new travelers to Rome to get their Vatican tickets early.
There's no problem at all getting Borghese Gallery tickets at the moment.