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John Jones' Tables http://hobbes.la.asu.edu/NFDB/ are my favorite. I think his data tables are the most complete online. Check for example in Klueners-Male tables for cubic fields of prime discriminant -3299, and you'll see that there are no results shown. However Jones' tables contains the 4 cubic fields with such discriminant.

Now about your question on the primes, as Ben mentioned p must be 1 mod 4 so the question has some hope. Even for p that are "allow" to be discriminants there might not exist a field of fix degree d of discriminant p. For example, class field theory tells you that there is a cubic field of discriminant p if and only if the 3-Sylow part of $Cl(\mathbb{Q}(\sqrt{p}))$ is non-trivial. In fact you can even tell how many of them there are! As an example of this we can conclude that there are no cubic fields of discriminant p=-3, even though p is a fundamental discriminant. A similar analysis can be done for quartic fields, and I think for them the behavior is related to the 2-Sylow part of $Cl(\mathbb{Q}(\sqrt{p}))$.

Also you might want to look at this paper of Jone's which I think is very close to your question http://hobbes.la.asu.edu/papers/OnePrimeJR.pdf

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John Jones' Tables http://hobbes.la.asu.edu/NFDB/ are my favorite. I think his data tables are the most complete online. Check for example in Klueners-Male tables for cubic fields of prime discriminant -3299, and you'll see that there are no results shown. However Jones' tables contains the 4 cubic fields with such discriminant.

Now about your question on the primes, as Ben mentioned p must be 1 mod 4 so the question has some hope. Even for p that are "allow" to be discriminants there might not exist a field of fix degree d of discriminant p. For example, class field theory tells you that there is a cubic field of discriminant p if and only if the 3-Sylow part of $Cl(\mathbb{Q}(\sqrt{p}))$ is non-trivial. In fact you can even tell how many of them there are! As an example of this we can conclude that there are no cubic fields of discriminant p=-3, even though p is a fundamental discriminant. A similar analysis can be done for quartic fields, and I think for them the behavior is related to the 2-Sylow part of $Cl(\mathbb{Q}(\sqrt{p}))$.