On average, how many landings does a student pilot need before their first solo?
My logbook shows 29 landings, with 11 hrs logged in an Aircraft Single Engine Land aircraft (ASEL) when my instructor stepped out and sent me off on my first solo flight.

I think those figures were typical for students at my flight school. It is probably also fairly typical for students at most §141 flight schools, such as mine. 

Now, there can be a significant difference between the experience of training under §61 and training at a §141 flight school. The schedule at §141 schools tends to be more fast-paced than what many students might experience doing §61 flight training. Since students must systematically complete the curriculum at §141 schools, a certain minimum level of experience will be in place before a student solos. However, this curriculum also lends itself to a student being ready to solo after a fairly defined period of time. A student training under §61 might take longer to be ready to solo due to a number of factors, including a possibly more relaxed training pace. On the other hand, §61 training can also give an instructor the freedom to solo a student after less time than might be typical.

I don't have any hard data outside of my own experience and what I have observed, but I would guess that a typical §141 student will solo after about 20-40 landings and 10-15 hours. I would guess that a typical §61 student will solo after about 20-60 landings and 10-25 hours.