Now, it's perfectly acceptable for a player to decide their character has one of these, and, indeed, many of them are 'free' proficiencies granted by social status or background rolls, but they aren't often that useful in game play. With that caveat, I will add this...I have DMed at least one situation in which someone having one of these skills completely changed the dynamics of the game.
Here are nine of these 'useless' non-weapon proficiencies. Each are from the "general" category. Remember that you don't need a proficiency to do something, you need a proficiency to do it at a professional level.
Baking: The character is familiar with the creation of breads, pastries and cakes, and can run brick ovens and similar kitchen implements.
Barristry: The character is proficient in laws, legal work and the manipulation of legal proceedings, and can represent himself or others in a court of law. Note that proficiency in Barristry doesn’t help in a courtroom using an alien legal system. In general, it is assumed this proficiency refers to the place where the character lives. Other places may be stipulated.
Candymaking: The character is proficient in working with scorching hot sugar and other substances to make various confections.
Cheesemaking: This proficiency allows the character who has it to expertly create cheese from the curds of soured milk. A proficiency check is required only when attempting to prepare a truly magnificent wheel of cheese as a special gift or for a special celebration.
Cobbling: The character can fashion and repair shoes, boots, and sandals.
Management: The character is skilled in directing people and delegating authority.
Maths: All characters with average or higher intelligence are assumed to be able to do basic sums, simple accounting and the like. The character with the maths proficiency exceeds this, with a working knowledge of algebra, geometry, trigonometry, descriptive statistics and generally any math not requiring calculus. Note that all characters with the Engineering Proficiency get maths for free.
Teamdriving: The character is skilled in driving large animals that pull large wagons, such as those used to move heavy loads, and draft animals for construction and logging. He can calm them from a panic, tell if they are likely to balk or spook and attempt to prevent it.
Throwing: Characters with this proficiency add 10' to each range category of thrown weapons, and increase the damage or the attack roll by +1 each time they throw a weapon. The player can elect to improve either the damage or attack roll, but the choice must be announced before the attack is made. For each character point spent on this proficiency (after its initial purchase) a character adds another 5' to thrown weapon ranges. For every 4 additional character points spent, another +1 on the damage or attack rolls is gained—this can be used as a +2 on one or the other, or split as a +1 to attack and +1 to damage. (Okay, this one was a trick to see if you were paying attention. YES, throwing is very useful.)
How these proficiencies changed an encounter:
Baking: The party found themselves in a Jack and The Beanstalk situation, and had approximately five minutes to hide themselves from a giant dog sniffing out the intruders on behalf of his giant master. The sympathetic (and also giant) kitchen maid was on their side. A character with the baking proficiency asked the DM how long it would be inside the dough of an (also giant) as yet uncooked loaf of bread going into the giant oven, and the DM replied "based on your knowledge of bread, you'd probably be safe for about 15 minutes inside the loaf."
Barristry: I think you can probably guess where this is going. An assassin skilled in all forms of the law found a technicality in her arrest and was set free by the government. In another instance, a thief managed to convince a judge that not only was he not guilty, he was being persecuted by the crown, and the judge threw his accuser in jail overnight.
Candymaking: While avoiding an attack in the kitchens of the castle, a character started a pot of butter and sugar simmering, then left it over the door. the trap set with this rudimentary caramel performed similarly to flaming oil, and did continuing rounds of damage to the (full plate armor wearing) attackers, as if the armor was being hit with Heat Metal. Since one of these knights was largely immune to magic, this was a big deal. The same character also made flaming sugar syrup missiles. (Candymaking is considered one of the most dangerous things you can do in the kitchen. I will do it only with much precaution, and I'm trained as a laboratory manager.)
Cobbling: Character (who also had tracking) examined the footprints of his quarry and determined he was a city guard because of the pattern of his boots, and that he had a slight limp to the left, then later identified him on patrol by his boots. "I also have cobbling, is there anything weird about his shoes?" asked the player.
Management: Character walked into a collection of kobolds, told them he was from the boss, and got them into a wagon by identifying who the leaders were, what they were doing, and who the weakest leader was. He then blew up the wagon. What would've been a bluff/lie check or a charisma check at a minus was done without a roll (except a reaction roll on the monsters.)
Maths: Player, who was not getting the fact that the trap involved prime numbers, asked the DM "I'm stumped, but (my character) has math, can I roll to see if she gets it?" (She did.)
Teamdriving: Character (who also had engineering) spooked a group of bison in just the right direction to make them take out a bridge being used by the enemy. He knew exactly where, when, and how, to do so. (Something that might otherwise have required animal handling, or a roll with a minus, was essentially an automatic success (the bison still had a reaction roll, but a very small chance of none spooking.)