We went to Rome for 6 days in early April. We picked up the free tickets for the Wednesday Papal audience from the Bishops' Office for United States Visitors to the Vatican which was located near the Trevi Fountain. Web site: http://www.pnac.org/general/visiting_vatican.htm#contact
We ordered the tickets on line and got a response the next day. Since after April 1, the audience is held in St. Peter's Square, I believe there is never any problem getting a ticket for the event. There were many many thousands of folding chairs set up for the masses of people in the square.
We got to St. Peter's square at about 9:30 for the 10:30 audience and got a good seat. Even those with worse seats had a good view, since there are 2 huge jumbotron TV screens set up for viewing the event.
The audience begins with the Pope being driven in his pope-mobile down and across the cordoned off lanes in the midst of the audience area, so almost everyone gets a good view of the Pope. Once the Pope gets to the covered area at the top of the steps, there is a short welcoming speech by one many priests/bishops/cardinals on the stand with the Pope and lots of individual groups in attendance were mentioned to the delight of those people. The Pope then read a lenghty document in Italian. This was followed by another priest reading another list of groups in attendance. At this point (probably 11:30), we decided to leave the audience along with quite a number of other people. I hope you do not go to hell for walking out on the pope, but by doing so, we were able to walk (15 minutes away) immediately into the Vatican museum. We then had lunch inside the museum also with no waiting.
So, my suggestion for seeing the Vatican Museum is to arrive about noon on Wednesdays, the day the Pope has his audience. Note, that I did walk to the entrance of the museum before the audience began to see if I could pre-buy our tickets for the museum and was denied. Once you buy the tickets, you must enter the museum.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.