I was looking for a reference which studies elliptic 3-folds (Their canonical bundle, second Chern class, singular fibers,...), similar to one for surfaces (Which is available in many books including Griffiths-Harris).
Rick Miranda's text (mentioned by Artie) describes how one can obtain an elliptic threefold from a Weierstrass equation and how the singular fibers behave in the particular model he constructed. (In the surface case there one has a unique smooth projective surface associated with a Weierstrass equation, in the threefold case this is false.)
There are several long texts by Noboru Nakayama on elliptic fibrations which contains several standard facts. (Mathscinet list 4 papers in total with the word "elliptic" in the title.)
Rania Wazir's text contains (besides the arithmetical part) a proof for the Shioda-Tate formula for elliptic threefolds.
There are texts by Grassi and Morrison which discuss the role of elliptic threefolds in string theory.
However, in the elliptic surface case you can obtain all the hodge numbers, chern classes etc. once you know a) the genus of the base curve and b) the degree of the discriminant (as a divisor on the base curve). Such a clean statement seems hard/impossible to obtain in the elliptic threefold case, unless you restrict yourself to certain classes of elliptic threefolds (e.g. one takes a Weierstrass model in some P^2-bundle and assumes that this is smooth).
Concerning the singular fibers: the discriminant is a (possible reducible) curve on the base surface. The fiber type over the generic point of an irreducible component is one of the fiber types from Kodaira's list. On each irreducible component there are finitely many points with a different type. Miranda calculated which possibilities there are for there special points, under the condition that you allow to replace your base surface with a birational one.