Factual error: Reid says that it's been speculated that Mary Jane Kelly was murdered in a flat rented by the killer - this is completely incorrect. She was murdered in her rooms, where she'd been living since early 1888.
No Way Out - S2-E13
Factual error: Gideon says that Frank is the most prolific serial killer ever, with 177 victims. This is incorrect - Pedro Lopez murdered 350+ young girls and Luis Garavito murdered 300+ plus girls.
Other mistake: In the opening scene, the UnSub is wearing breathing apparatus, however his smoke hood is under his face mask. Wearing the hood under the mask would make it impossible to get a face seal, and as such would let smoke into the mask.
Character mistake: Dr. Reid calls Penelope "Kristy" on a phone call.
Suggested correction: He calls her "Garcia."