Yes, they can be common. Many situations like being in a long line, in a vehicle during traffic, or at an event where leaving briefly is inconvenient can lead to such situations for girls. It's a normal part of life experiences.
A girl was in a meeting that went on much longer than expected. She had already had a large coffee before the meeting. As the meeting dragged on, she could feel the pressure building. She tried to sit still and look composed, but inside she was really desperate. She was relieved when the meeting finally ended so she could go to the bathroom.
There was a girl at a concert. She drank a lot of soda before it started. As the concert went on, she started to feel the need to pee desperately. The crowd was so thick that she could hardly move. She tried to focus on the music to forget about it but it didn't work. She had to push through the people, saying 'excuse me' over and over again until she reached the restrooms at the back of the venue.
They can be common. Moms are frequently on the go, running errands, taking kids to activities, etc. In these situations, it can be hard to find a convenient time to pee. Say a mom is at a park with her kids. There may not be a close bathroom, and she can't just leave the kids alone to go, so she ends up getting desperate.
There was a student who was in the middle of an important exam. The exam rules didn't allow leaving the room. As time passed, the urge to pee became intense. He held it throughout the whole exam, constantly shifting in his seat. After the exam, he rushed to the bathroom. It was a real test of his willpower.
A 'pee desperation fanfic' is a type of fan - created fictional work that often focuses on the theme of a character experiencing extreme urgency to urinate. It might be part of a niche genre within fanfiction communities, but it's not very common in mainstream literature.
At school or university, a girl might be in a class that runs for a long time without a break. If she has had a lot to drink before class, she could be in a pee desperation situation. Another common scenario is when she is out hiking. There may not be proper restroom facilities around and she has to hold it until she finds a suitable place. Moreover, during a movie in a theater, especially if it's a long movie, she might get desperate if she had a big drink before.
Once, there was a student named Tom. He had a big glass of water right before a long class. Halfway through the class, he started feeling the urge to pee desperately. He fidgeted in his seat, but the teacher was strict and wouldn't let anyone leave without a good reason. He held it until the end of the class and rushed to the bathroom.
Here's another one. A hiker got lost in the mountains. He was so thirsty that he drank from a stream. But soon he realized that there were no proper facilities around. As the day went on, the pee desperation grew. He had to keep moving to find his way back and all the while he was really struggling to hold it. Eventually, he found a secluded spot and relieved himself.
In my middle school, a girl was in a long assembly. She had a weak bladder. As the assembly dragged on, she started to look really uncomfortable. Her friends noticed but there was not much they could do. She kept crossing and uncrossing her legs. Finally, when the assembly ended, she ran as fast as she could to the bathroom. It was quite an embarrassing situation for her but luckily she reached in time.