The following classes are of a slightly different flavour because they depend on the additional choice of a connection.

Assume that $E\to B$ carries a flat connection $\nabla$. Then the Kamber-Tondeur classes are obstructions against the existence of a $\nabla$-parallel metric on $E$. In the case of a complex line bundle, the first Kamber-Tondeur class is the only obstruction.

The Cheeger-Simons differential characters of a vector bundle $E\to B$ with connection $\nabla$ are obstructions against a parallel trivialisation. For a complex line bundle, the first Cheeger-Simons class is the only obstruction (in fact, this class classifies complex line bundles with connections).

Note that the Kamber-Tondeur classes can be interpreted as the imaginary parts of the Cheeger-Simons differential characters.