Alan Rickman turned down the role of the Sheriff twice before he was told he could more or less have carte blanche with his interpretation of the character.
Alan Rickman's famous line of "Locksley, I'm going to cut your heart out....with a spoon" was an improvisation. He had found his character's dialogue to be bland and one-dimensional, so was always looking at ways to make it more interesting and amusing. Interestingly, Costner does not break character during this scene despite the improvisation and the inclusion of it resulted in another scene providing explanation.
Christian Slater ad-libbed Will Scarlett's line of "Fuck me, he cleared it!", after Robin and Azeem are catapulted over the castle wall. Director Kevin Reynolds kept it in the movie because it was funny, despite the historical inaccuracy.
In the scene where the Sheriff of Nottingham is kneeling down in front of a little girl and holding her arms, while speaking to her about having a sad childhood, took a few extra takes. Before the scene was shot, the child actress was introduced to Alan Rickman so she would feel at ease with him. Alan was known to love children and work well with them. So he wouldn't frighten her, he would make silly faces at her and smile while saying his lines. Which caused her to keep giggling during the filming.