With DSA / ECDSA using a fixed $k$ means the private key can be trivially easily calculated from two different signatures. To remedy this [RFC6979 Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA)](https://tools.ietf.org/html/rfc6979) was developed.

[OAEP](https://en.wikipedia.org/wiki/Optimal_asymmetric_encryption_padding) / [PSS](https://en.wikipedia.org/wiki/Probabilistic_signature_scheme), as described in PKCS#1 v2.0+, used randomized values as well.

For PSS I guess it isn't a huge problem since the only randomized element is the salt and the salt has a variable length with 0 being a valid length.

For OAEP my assumption is that the only issue with a fixed `seed` is that if the same message were encrypted twice you'd be see the same ciphertext each time. Or is there another more severe issue that the randomized `seed` protects against?

I note that the original encryption described in PKCS1 employees randomized padding as well.