This is a common and powerful idiom used to criticize. Its connotation is strongly negative.
In Arguments & Debates: It's used to dismantle an opponent's argument by pointing out its many logical fallacies and inconsistencies. “Your theory is `破绽百出`.”
In Business: A business plan, proposal, or financial report that is poorly researched and full of errors might be described this way.
In Daily Life: People use it to call out a weak excuse or an obvious lie. If a child tells a convoluted story about why they didn't do their homework, a parent might say the story is `破绽百出`.
Media & Reviews: Film critics might use it to describe a movie with a nonsensical plot. News commentators use it to critique flawed government policies.
Formality: It can be used in both formal writing (e.g., academic papers, news reports) and informal conversation, but it always carries a serious, critical tone.