One great 'chicago pd fanfic' is 'The Blue Line Bonds'. It delves deep into the relationships between the characters outside of their regular work scenarios. The character development is on point and it gives a fresh perspective on the show's dynamics.
One possible fanfiction could be where Lindsay is injured during a dangerous undercover operation. Maybe she gets stabbed while trying to infiltrate a criminal gang. Her partner then has to fight to get her to safety in time and she has to battle her injuries while also dealing with the emotional trauma of the situation.
One great thing about Chicago PD and Law & Order: SVU crossover fanfiction is the unique character interactions. Writers can bring together the tough cops from Chicago PD and the specialized detectives from SVU. For example, they might create a scenario where a case starts in Chicago but has ties to New York, leading to a joint investigation. This allows for some really interesting dynamics as the characters from different shows with different methods have to work together.
The 'roommates' trope can also be found in Chicago PD gay fanfic. Imagine two characters who end up living together for some reason, and as they share a living space, they start to notice things about each other that they hadn't before, leading to a romantic relationship. And of course, the 'secret relationship' trope is popular. Given the nature of their jobs in the show, two male characters might keep their relationship a secret from their colleagues at first.
Often, you'll see themes like the OC having a special skill or background that helps Jay and the team solve cases. There's also the theme of the OC facing personal struggles while being part of the 'chicago pd' environment. And of course, the theme of friendship evolving into something more between Jay and the OC is quite prevalent. In many fanfics, the OC might be a new recruit, and the story shows how they grow and interact with Jay in that context.
One idea is that Lindsay gets hurt by a booby - trap. She's exploring a crime scene and accidentally triggers a hidden trap. It injures her leg badly. She then has to rely on her colleagues to carry on the investigation while she recovers. This could also lead to some interesting character development as she has to deal with being incapacitated for a while.