I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that: 


i) they vanish if the bundle of unit spheres $S(E)$ of the vector bundle $E$ (viewed as a real vector bundle if $E$ is complex) is stably fiberwise-homotopically trivial. 


ii) the are well behaved under direct image(umkehr) homomorphism, for bundles oriented over the corresponding cohomology theory e.g., skew functoriality etc... 


Other than Stiefel-Whitney and Wu classes in ordinary cohomology I know only Bott's cannibalistic classes in $K$-theory. But I don't know how Bott classes behave with regard to ii). Moreover my feeling is that, precisely because of ii), the right place for the characteristic classes which I need should be the complex cobordism.


Any reference to related topics would be more than welcome. 



Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles
An evident general construction is to take any multiplicative cohomology theory $E^\*$ and a cohomology operation $\psi$ in that theory, say taking $E^q$ to $E^{q+n}$. For an $E$-oriented real $q$-bundle $\xi$ over $X$, $\theta^{-1}(\psi(\mu))$ gives the value of a characteristic class, where $\mu$ is the Thom class of $\xi$ and $\theta$ is the Thom isomorphism. For stable $\psi$, defined for all $q$, this will give a characteristic class on all $E^\*$-oriented bundles of the sort wanted, modulo precision about condition (ii). [Senior moment nonsense eliminated]. You are studying the J-map $BO\to BF$ (or $BU \to BSF$), where $BF$ classifies stable spherical fibrations (oriented for $BSF$). A lot more is known than Adams knew. In particular, he didn't yet have the Adams conjecture. Rationally, $BF$ is a point. At an odd prime $p$, $BF$ factors as $BJ\times BCokerJ$, and at $2$ there is a non-split fiber sequence $BCokerJ \to BSF \to BJ$. The $J$-map at $p$ is best thought of as a map $BSpin\to BSF$ ($BO\simeq BSpin$ at $p>2$). By the Atiyah-Bott-Shapiro orientation, the $J$-map $BSpin\to BSF$ factors through the classifying space $B(SF;kO)$ for $kO$-oriented spherical fibrations and, at any $p$, $B(SF;kO)$ splits as $BSpin\times BCokerJ$ (but BSpin is seen in
two pieces, one carrying the Wu classes, the other the rest of the Adams splitting). The intuition is that $BCokerJ$ and thus the unknown parts of the stable homotopy groups of spheres can be ignored, leaving the focus on the quite computable composite $BSpin \to BSF \to BJ$. This is too fast, and details are in Chapter V of $E\_{\infty}$ ring spaces and $E\_{\infty}$ ring spectra''. In ordinary mod $p$ cohomology, calculations are thoroughly understood but don't shed light on your question. They are also understood for $K$-theory, by work of Hodgkin and Snaith, and here the intuition that $Coker J$ can be ignored is made precise by Hodgkin's result that $\tilde K(BCokerJ)=0$.


