Signed quantities are a good default, and are easier to deal with when doing subtractions and mixing integers of different widths. (And integers includes pointers here, so it's very hard to not have different widths).
However unsigned integers are still very useful, I'd say essential, in low-level programming. For example when doing buffer management and memory allocation.
- bitwise operations
- modular arithmetic implemented with just ++, -- (ringbuffers, e.g TCP sequence numbers)
- using the full range of a 8-bit, 16-bit, 32-bit datatype (quite common)
- splitting a positive quantity into two smaller quantities, e.g. using a 16-bit index as 8-bit major index plus 8-bit minor index.
Don't forget that the signed vs unsigned integer is in some sense an artificial distinction. Machines have you put the distinction in the CPU instructions themselves, they don't track a "signed" property as part of values. And it can make sense to use the same value in different ways. However, C and many other languages decided to put a tag on the type, so operator syntax can be agnostic to signedness, and the compiler will choose the appropriate CPU instruction.