How to store carrots
Best place: the fridge. USDA gives carrots a fridge time (2 to 3 weeks, from the package date) and no pantry time, so they belong in the fridge as soon as you get home. The freezer gives 10 to 12 months.

Where these times come from. USDA FoodKeeper has no entry named "carrots". The 2 kinds on this page are the USDA entries that cover carrots, copied without change from Carrots, parsnips and Baby carrots. Each row in the table is one USDA entry; pick the one that matches what you have.
No guideline
2 to 3 weeks
from the package date
10 to 12 months
best quality within
USDA notes for carrots
- baby carrots: below 75 °F
- baby carrots: Blanch before freezing.
How to tell carrots have gone bad
General signs for produce: fresh vegetables, the group USDA lists carrots under.
- Slimy, wet leaves on greens that were crisp before
- Soft or mushy spots on vegetables that were firm
- Sprouting or wrinkled skin on potatoes and onions
- Mould, often white or fuzzy, near the stem or cut end
- Sour or off smell replacing a vegetable's usual earthy scent
More on produce: fresh vegetables. The times above are quality guidelines, not safety limits; food that shows these signs goes out regardless of the date.
Storage habits that make the difference
Store leafy greens loosely wrapped in the crisper drawer, ideally with a paper towel to absorb extra moisture. Keep potatoes, onions and garlic in a cool, dark, dry place outside the fridge, and store potatoes away from onions since they spoil each other faster. Trim and wash vegetables just before cooking.
How to store other produce: fresh vegetables
Source: USDA FSIS FoodKeeper dataset (public domain). How we decode the data.