Combining game watching and daytime drinking with learning this weekend (three things I like very much), I was reading up on NFL rules this weekend, and learned that there are two ways to score that remain in the rule books but NEVER happen. 

One is the drop kick for an extra point, which is, according to NFL rules, “a kick by a kicker who drops the ball and kicks it as, or immediately after, it touches the ground.” The play dates back to the era (pre 1934) when a football was more round and easier to kick in a predictable direction off a bounce or ones shins. In 2006 Doug Flutie, playing for the Patriots, converted the first successful drop kick in an NFL game since 1941: As covered in ESPN:

I think Doug deserves it,“ said usually dour Patriots coach Bill Belichick, who broke into a wide smile when his sprightly quarterback split the uprights off one bounce. "He is a guy that adds a lot to this game of football, has added a lot through his great career – running, passing and now kicking.

"He’s got a skill and we got a chance to let him use it, and I am happy for him. First time since ‘41,” said Belichick, a football historian who last month brought out a leather helmet in his media session. “It might be 60 years again, too.”

The other is the fair catch kick, where a player makes a fair catch of an opponent’s kick, and then immediately kicks a field goal from the spot of the catch for three points. Its distinguishing marks are that it’s the only type of kick where a kicking team can score a field goal, and onside kicks are not permitted. The stars need to align in an uncommon way to make a fair catch kick advantageous: the fair catch needs to be made in the first place, at a point close enough to make a field goal plausible; there needs to be insufficient time to run more than one play from the line of scrimmage;  and the score needs to make three points significant.

The last successful fair kick catch was completed in NFL was completed by Ray Wersching in 1976 for the San Diego Chargers against the Buffalo Bills. It seems (according to Wikipedia) to be attempted most often with no time remaining on the clock to run a play.