Applications for one-way-functions in cryptography
Hash-collisions may happen in rare cases, but are mostly disregarded here.
A quick way to ensure integrity of data is to compare two hashes, where one is a previously calculated hash and the other is the newly calculated hash of the data, which is presumed to be unmodified. If the previous hash matches with the new one, then the data has in fact not been changed.
Comparing two hashes is computationally very efficient compared to "bit-by-bit" comparison of large data.
Downloadable files often have a checksum (usually SHA256) for that same file published as well. This way you can also compute the checksum after downloading the file. If the checksums match then you know that the file hasn't been modified, i.e. adding backdoors, viruses, worms, etc.
I won't go into detail here, because you mainly asked for other applications than password-hashing. The main advantage of one-way-functions concerning passwords is obviously, that you don't have to store the passwords as plaintext and still can authenticate users.
The function of digital signatures is basically the same as signatures on paper with ink. They ensure the authenticity of the source not of the file. They are very commonly used in Emails, this way the receiver of a message can verify that the sender is in fact the person who wrote the Email.
A popular example of proof-of-work is the mining of bitcoins, where miners have to calculate a certain hash-value. This way it's relatively easy to verify a certain value (in the example of bitcoin-mining you "verify" the integrity of the decentralized nodes in the P2P bitcoin network) but very hard to do the same work for a possible attacker.
CSPRNG's have a relatively wide use in cryptography, i.e:
CSPRNGs in contrast to "normal" PSNGs must have the requirement of being one-way-functions (which again is not yet proven if such generators exist).
A KDF is used to retrive several (at least one) secret keys from a master-secret-key. Keyed cryptographic hash functions are popular examples of pseudorandom functions used for key derivation.