Lampshaded and averted in Shrek, especially in a scene where Robin Hood and his Merry Men try to "rescue" Fiona from the ogre they believe has kidnapped her, only to have her rebuff him and beat up all his men in a combination of styles from Xena: Warrior Princess and The Matrix.
And in Shrek Forever After, where in an alternate universe where Shrek was never born and never came for her, Fiona eventually decided to rescue herself.
Played straight at first but later subverted with Fiona in Shrek 4-D, who at first is helpless after Thelonius kidnaps her, but eventually gains the upper hand and beats him.
Hercules: Aren't you a damsel in distress? Megara: I'm a damsel... I'm in distress... I can handle this. Have a nice day.
A more recent straight example would be Kida from 2001's Atlantis: The Lost Empire, who spends the last third of the film crystallized by the villain, and that her boyfriend and his teammates actually had to rescue her and change her back.
Subverted in Titan A.E., when Akima is jettisoned into space, captured, and held to be sold into slavery. The rest of the crew undergoes a makeshift rescue operation, only to find out that she successfully knocked out all of her captors and is patiently waiting to be picked up.
Played with in Happily Never After, in which The Prince (whose name is revealed to be Humperdink) is searching for one of these (or a lady in waiting or whatever else is a typical princess) and sounds excited that Ella could be one of those things. When he asks if she's a damsel in distress, her response is "I will be. Kind of. At midnight". To say the least, Ella does more ass-kicking than servant boy Rick or Humperdink.
Straight example in Hoodwinked, where Red Puckett is Bound and Gagged and loaded into a tramway cabin filled with dynamite, and it's up to Granny to rescue her.
Michelle, the little badger from Once Upon a Forest, inhales a poisonous gas and falls ill. Thus, it's up to her friends Edgar, Russel, and Abigail to find the herbs to heal her. Her damsel status stands in a bit of contrast to Abigail, an Action Girl who fights an owl while trying to save her.
Odette from The Swan Princess is under an enchantment by Rothbart that causes her to turn into a swan when moonlight leaves the lake by his castle in the morning and she needs Derek to break the enchantment, but she also turns down Rothbart's marriage proposals without a hint of remorse or fear even though she knows he's a powerful sorcerer and he killed her father. Also, she doesn't exactly wait for Derek to show up and does everything she can to try and let him know where she is and what he has to do.
The Fuma Conspiracy: Murasaki, who twice found herself the target of kidnapping and/or traps. The first was when she was captured by a Fuma Ninja during her wedding, and demanded the MacGuffin in exchange for her life. The demand motivated Goemon and company to steal the vase and deliver it to the Fuma.
Lupin III: The Columbus Files shows an example of Fujiko being a Damsel in Distressand a badass, but without being a Badass in Distress. During the opening, she loses her memory, and her entire personality changes. She's terrified of nearly everything, and everyone is a stranger to her. When she contributed to part of her rescue via pure Muscle Memory, that also terrifies her.