While Dr. No was considered a good film, From Russia with Love is where things started to grow for James Bond. It was a stronger film and introduced many of the things we would think of with the series. It was an even greater hit and is among the fans often (and by Sean Connery himself) considered to be the greatest Bond of them all and put the series on the map. Goldfinger is another strong contender and was the start for the Bond-fever. The Spy Who Loved Me is often considered the film where Roger Moore grew the beard, but many also consider him to have lost it soon afterwards.
Harry Potter and the Prisoner of Azkaban; while it was a more contentious example with the fandom due to being a Pragmatic Adaptation, it was the best reviewed Harry Potter movie (up until Deathly Hallows Part 2 came out), and it was regarded by critics as an improvement over the first two films. The remainder of the films would actually go on to mimic Azkaban's tone and pragmatic cuts.
Marvel One-Shots kicked off with a short comprising of nothing more than a clip from The Incredible Hulk bookended by clips of SHIELD agents conversing at a table. The following shorts rely more heavily on original footage, and boast more interesting plotlines. And now Marvel's thinking about using them to test new characters and see if they're received well enough to make the jump to the Marvel Cinematic Universe movies.
The Star Trek film series itself grew the beard between its first and second installments. The Motion Picture was devised as a grand meditation on man's place in the universe, but ended up as a shallow light show filled with wooden actors playing cardboard characters. In contrast, The Wrath of Khan was a swashbuckling adventure in space that dealt far more successfully with weightier themes than its predecessor. The film series went through the exact same cycle with The Final Frontier and The Undiscovered Country; the former film was a self-conscious epic that fell on its face, whereas the sequel was superficially a detective mystery, but one that dealt with dark and complex themes (in the form of a political fable about the end of the Cold War) in an entertaining way. The cycle repeated itself once again with the Darker and EdgierNemesis, which was such a disaster that the series needed a reboot.
The Three Stooges started growing their beards when they broke away from Ted Healy. Although their first short ("Woman Haters") was a bit of a misfire, the beard was full in their next effort, "Punch Drunks". The beard was trimmed when Curly's declining health hampered several promising shorts, but the beard grew back, albeit slightly less fully, when Shemp replaced Curly. Most fans would agree, though, that the beard fell out completely and took the whole jawline with it when Shemp died and was replaced by Joe Besser.