Do deterministic secret sharing schemes exist? Are there algorithms for secure secret sharing such that the algorithm depends solely on the value being secured, relying on no randomness in its calculations.

If there aren't, is such an algorithm an impossibility? If there are, what are some examples?

I ask because I've come up with a secret sharing scheme (which is probably littered with flaws ;-)), and I'd like try and prove certain properties about it. If deterministic secret sharing schemes are a no-go, I'd very much appreciate a heads up.

I'm not entirely certain this question is well defined. If it is not, I would greatly appreciate any information on why it is ill defined and how to make it more rigorous if possible.
Any deterministic secret sharing scheme as in the question has the property that any participant can run the deterministic algorithm for a guess of the shared secret, and eliminate the guess if the share that the algorithm deterministically assigns him/her does not match his/her share. This implies that some information about the secret is leaked in his/her share, or that his/her share is pointless (in the sense that his/her share can be removed from any set of shares that allows to reconstruct the secret, while keeping that ability).

Likely that's why there is no seldom such beast in the literature. For a counterexample to this assertion, see that other answer.