Bucket sort works best in these situations:

1. Even Distribution: It shines when the data is evenly spread out across a range. This means that when the items are balanced, it can sort them quickly, getting close to $O(n)$ in average time.

2. Small Range: It's great for sorting numbers that fit within a small range. For example, if you’re sorting numbers from $0$ to $100$, it uses less space and is more efficient.

3. Big Data Sets: It works well when you're dealing with large amounts of data that have many repeated values. This is because it cuts down on the number of comparisons needed. For example, if you have 1,000,000 items and 100 unique values, the average performance is around $O(n + k)$.