10 Dumbest Things In Star Trek: Deep Space Nine
10. Odo's Combadge
Let's start with a classic and a rabbit hole that might never end.
While Deep Space Nine's head of station security becomes a clear security risk himself by the beginning of season three, there are character relationships which help to iron out that little wrinkle. The bigger issue is his shapeshifting ability. From the very start it's clear that his appearance is the best approximation he can manage to humanoid. This is backed up by the fact that his hair is based on that of scientist Mora Pol. By the the alternate future in Children of Time, his ability to retain form and define features has become substantially better.
This begs the question of how Odo's combadge can exist? There are two options here. One is that it is a combadge and is somehow hidden within each form he takes, whether it's a rodent in A Man Alone or a spinning top in Shadowplay. However that doesn't explain how it is seen to dissolve when Odo changes form or when moving around in a gelatinous state such as escaping from the Cardassian storage locker in Emissary.
Given the intricate circuitry taken to construct such a subspace communications device, it seems very unlikely Odo could create its functionality in such minute detail given that he finds noses rather difficult. Take into account as well that he not only maintains humanoid form but also the clothing too which is in itself textured and moves as he moves. That's a lot to be dealing with when you're taking on solid form.