In terms of a play, skills/influences refresh at the end of an act, but not the end of a scene.
You will always get something when you spend investigative skill points. It might not be useful, but it won't be false. Also, it's important to remember that your skill level remains the same whether or not you spend points, and so you will always find the appropriate clues for your character. Points get you optional extra information or the chance to dominate the scene or hog the spotlight. Because Trail of Cthulhu has a very narrativist structure, you will always get something that lets you move to the next scene, whether or not you spend points.
Eg Mattias used Intimidate to get some extra information about the crime scene. He would probably have got that info when visiting the scene anyway, but can now prepare in advance.