TITLE: A dictionary of Characteristic classes and obstructions
QUESTION [57 upvotes]: I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. 
In an effort to get a hold of characteristic classes I had the idea of compiling a short dictionary relating characteristic classes to their obstructions. Unfortunately I didn't find anything of this sort on the web.
It could be a nice thing If we could compile such a list here. 

Let $E \to B$ be a real vector bundle over a compact manifold (for simplicity):

Euler class ($E$ orientable): $E \to B$ has a nowhere vanishing section $ \implies e(E)=0 $.
Stiefel-Whitney classes: 

$w_1(E)=w_1(\det E)=0 \iff E$ orientable. 
$w_1(E) =w_2(E) = 0 \iff E$ has spin structure.
$E$ has a trivial subbundle of rank $m$ $\implies$ $w_k=0$ for all $k>rank(E)-m$.
$E$ orientable $\implies$ $w_{top} (E) = e(E) \text{ mod 2}$

Pontryagin classes:

For $E$ spin vector bundle: $\frac{1}{2} p_1(E)=0 \iff E$ has string structure.
For $E$ string vector bundle: $\frac{1}{6}p_2(E)= 0 \iff E$ has 5-brane structure.
If $rank(E)$ is even: $e(E) \cup e(E) = p_{top}(E)$ 

Chern classes: Suppose $E \to B$ is now a complex vector bundle.

$E$ has a trivial complex subbundle (or is it quotient bundle here?) of rank $m$ $\implies$ $c_k=0$ for all $k>rank(E)-m$.
$c_i(E)=w_{2i}(E_{\mathbb{R}}) \text{ mod 2}$.
$c_1(E) = c_1(\wedge^{top} E) = 0 \iff E$ has reduction of structure group to $SU$. I read in several places that this has something to do with the possible number of linearly independent parallel spinors - notice $w_2( E_{ \mathbb{R}}) = c_1(E) = 0$ so $E$ is spin in particular.
$c_{top}(E)=e(E_{\mathbb{R}})$

Todd class: ?
Chern character: ?
Wu class: ?

Additions and corrections are welcome.

REPLY [12 votes]: 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.