I don't suppose you could…?
What it really means
A famously indirect British request. Starting with "I don't suppose" politely expects the answer might be no, which makes it easier for both sides. The speaker genuinely wants the favour.
Context, tips & common mistakes
"I don't suppose you could…?" makes a request through the negative — 'I don't suppose you could swap shifts?' — pre-packing the refusal so declining costs nothing: 'sorry, no' simply confirms what the asker already 'supposed'. It is the politest way English offers to ask for something you suspect is unavailable, and its siblings include 'you wouldn't happen to…' and 'I don't imagine there's any chance…'.
Two practical notes. Answering: despite the negative grammar, respond to the request, not the syntax — 'actually, yes!' is the happy answer and confuses no one. Asking: this construction shines with strangers and long-shot favours ('I don't suppose this seat's free?'), where it reads as considerate rather than pessimistic; among friends it can sound oddly formal, so keep it for when the social distance justifies the padding.