By default, I ask for permission to use people's personal stuff, and on rare occasions especially with close pals it's a habit not that I truly value their opinion in my taking their property.
When it comes to a spouse, I would not just ask to use their computer or phone by default, I am actually careful of intruding on privacy.
Your spouse is still a person and still has a right over his privacy. Asking to use their stuff when necessary wouldn't kill you. However, observe their pattern of response to your request. Do they hesitate? Are they sneaky with files, documents or chats? Observations are key, but sometimes, if he's very respectful of you, he won't leave trash lying around.
NB: Everybody has trash, including yourself.