My inventory shows that I have lots of tools, but when I go to an investigation I don't see them listed..

When you play an investigation some tools are listed at the bottom left and right of the play field (like magnifying glass, the timer etc.) and you can use them in the investigation.

Others show up when you chose between investigation and match-3 (also left and right) and must be put to use beforehand (like microscope or the colour-bubble).

But some tools like anti-bomb or the needle for the balloons seem only to show up, when you need them. There seems no possibility to activate them yourself.