I'm also curious in how these systems prevent grinding. What's to prevent the characters from rolling climb rope 20 times a day until they get the skill where they want? In Mouse Guard (and possibly Burning Wheel) the actual skill checks are a finite resource and you only get so many per day. How else can the game reward skill use without becoming a boot camp simulator?
Mouse Guard
Mouse Guard has explicit GM's Turns and Players' Turns. To advance a skill, you need a mix of successful and failed tests, and failed tests have consequences in the form of conditions (e.g., Sick, Angry, Injured) or twists (complications). For the GM's Turn, the GM dictates what skills will help surpass the obstacles in question. There's some negotiation in this, but the subset of useful skills for a given obstacle is limited. For example, you're unlikely to break out Fighter to settle an otherwise amicable trade dispute.
In the Players' Turn, players get one test (roll of the dice) for free and can earn "checks" in the GM's Turn for additional tests in the Players' Turn by hindering themselves and breaking ties against themselves. They can use these tests for pretty much any situation they want to get into that narratively makes sense based on where the GM's Turn ended. So, if a player really wants to work on their mouse's Fighter skill, they can go get in a fight. However, there's a finite number of checks they can earn, and conditions and twists that come from failed tests will make things difficult for them to accomplish their mission. In particular, the Sick and Injured conditions reduce the dice they roll, making it more likely for them to keep failing.
Personally, if the player writes a Belief like, "I will be the best fighter in the Territories", then I would be totally down with them getting into fights and advancing their Fighter skill to work on that belief. My goal, as the GM, is to provide conflict for that Belief and their short-term, usually-mission-related Goal to get the player to roleplay out choosing between their desires and the work of being in the Guard.
Burning Wheel
There are three rules in Burning Wheel that keep this from being a problem.
Roll Dice or Say “Yes”
If the GM can't come up with an interesting consequence for failure, given the players task (what they want to do) and intent (what skill they want to use), then the GM can just say "Yes" and give the player their intent instead of rolling the dice. In play, that means every roll has a consequence and you only roll when it's important. If you're at home climbing a rope, you're not rolling Climbing—you're not rolling at all. Even if you get in a fight or argument, you can only earn one test for a given scene (even if you use Sword or Persuasion the entire time).
Let it Ride
The result from a roll stands ("rides") until the situation has changed dramatically. If you roll and fail, you don't get to reroll. You've failed; move on to something else. You only test the skill again until you get in another situation where it's relevant.
Practice
Just because you don't get a Climbing roll for practicing your climbing in a safe environment, you can earn still tests this way. However, you need to work on it for hours a day for anywhere from a month to a year at a time, depending on the type of skill and what kind of test you want. (The time doesn't have to be contiguous, but you do need to log them all.)
For example, Climbing is Physical, so to get a routine Climbing test you would need to log two hours of climbing for a month. If you wanted a difficult test for Carpentry, it's a Craftsman skill, so you need to spend eight hours a day for a year to earn a test. Generally, Practice only comes up as downtime between sessions, e.g., "We're going to fast-forward a month, tell me what you're practicing."
