Let $M$ be a closed oriented manifold, an oriented foliation $F$ is said non-trivial, if $F$ is not fibration of $M$, i.e. there does not exist a closed manifold $B$, such that $M\overset{F}{\to} B$.
Q Is there any book or anything giving the explicit construct such non-trivial foliation.
PS: What I knew is just the Reeb foliation on $S^3$ and $T^n$ with a family of dense lines? Is there any other example (excluding the product of two manifold)?
Foliations of surfaces only exist on surfaces of Euler characteristic 0 and they are all built from suspension foliations and Reeb components. (See the book by Hector and Hirsch).
On 3-manifolds there exist many more foliations. Gabai has proved that for every homology class $h\in H_2$ one finds a taut foliation (without Reeb components) with the surface $H$ (that represents $h$) as a leaf. On the other hand, $H$ can be a fiber only if the Poincaré-dual cohomology class $h\in H^1$ has a finitely generated kernel when seen as a homomorphism $h\colon\pi_1\to Z$. (Stallings Theorem)
Gabai‘s construction is by induction: he cuts the manifold along the surface $H$, shows that this process can be continued until one arrives at a product (bordered surface x interval) (which can obviously be foliated) and then reverses the proces while extending the foliation.
So, if you want explicit examples, one might try to start with a product (bordered surface x interval), and follow Gabai’s paper to glue it along some subsets of the boundary to get nontrivial foliations according to Gabai‘s construction.
If you prefer complicated 1-dimensional foliations, you may look at Vogt‘s paper from Publ. IHES 1988, where he constructs a circle foliation of $R^3$. This can obviously not be a fibration.
