Normal invariants

Unfortunately I do not know any good references for detailed computations, but let me point out that the rational picture is rather simple: we have $$G/O \sim_{\mathbb Q} BO \sim_{\mathbb Q} = \prod_{i \geq 1} K(\mathbb Q,4i)$$ and thus $$ [M,G/O] \sim_{\mathbb Q} \prod_{i \geq 1} H^{4i}(M;\mathbb Q)$$ In particular, $\mathcal N(S^n)$ is finite if $n \not\equiv 0 \ (\text{mod} \ 4)$, and infinite cyclic if $n \equiv 0 \ (\text{mod} \ 4)$, in line with the analogous statement for the $L$-groups of $\mathbb Z$.

Many examples of computations of $[M,G/O]$ appear in papers which apply surgery theory. Here are some examples:

• Brumfiel did the complex projective spaces $\mathbb{C}P^n$.
• Land did the complex projective space $\mathbb{C}P^2$.
• Kirby-Siebenmann did high-dimensional tori in Appendix V.B.
• Crowley did products of spheres $S^p \times S^q$ for $p,q \geq 2$ and $p+q \geq 5$.

If you want specifically low-dimensional calculations, the Kirby-Taylor article A survey of 4-manifolds through the eyes of surgery does this for 4-manifolds. The discussion highlights the difference between the topological and smooth (= PL in this dimension) cases.