Repeatedly when Meg speaks Sam and Dean say, "Shut up Meg!" This is a play on a running joke from Family Guy (1999) in which the father Peter continually says that whenever his daughter, Meg, says something he finds annoying.
When Sam and Dean find Meg tied up in the bathroom, she asks, "Aren't you a little short for a storm trooper?" That's what Princess Leia asks Luke when he finds her locked up aboard the Death Star in Star Wars: Episode IV - A New Hope (1977).
Sam introduces himself and Dean as Special Agents Tandy and Lynne. This is likely a reference to the musicians Richard Tandy and Jeff Lynne, both members of the rock group Electric Light Orchestra.
Meg calls Castiel "Clarence." Clarence is the angel trying to get his wings in It's a Wonderful Life (1946).
The Spear of Destiny that Dean finds in the vault of the Men of Letters is also known as the Holy Lance, the Holy Spear, Lance of Longinus, and Spear of Longinus - that is, the spear that pierced Jesus in the Gospel of John.