Heap Sort, Quick Sort, and Merge Sort are three important ways to sort items. They each have their own features and can work differently based on the situation.

Time Complexity:

• Quick Sort is usually the fastest choice, working at an average time of $O(n \log n)$. This makes it great for most everyday tasks. But, if the data is already in order or almost in order, it can slow down to $O(n^2)$.

• Merge Sort also works at $O(n \log n)$ for both average and worst-case situations. This means it runs consistently well, no matter how the data is arranged.

• Heap Sort also stays at $O(n \log n)$ but is not as effective in real-life use because it can take more time to run because of noticeable overhead.

Space Complexity:

• Merge Sort needs extra space, about $O(n)$, to do its job. This can be tough to handle if you're low on memory.

• Quick Sort is much better in this area. Since it sorts items in place, it only needs about $O(\log n)$ extra space, making it a good option.

• Heap Sort is similar to Quick Sort, as it also sorts items in place, needing just $O(1)$ extra space.

Stability:

• Merge Sort is stable, which means that when it sorts, it keeps equal items in their original order. This is important for certain tasks.

• Quick Sort and Heap Sort are not stable, which can be a problem when the order of equal items is important.

In summary, while Heap Sort has good and steady performance, it often doesn't match up to Quick Sort and Merge Sort in real-world use. Choosing the right sorting method depends on what you're trying to do and the specific needs you have. Each sorting method has its strengths and can be useful in different situations.