Consider the sphere $S^n = \{x \in \mathbb{R}^{n+1}; \| x \| =1 \}$. Is true of false: the normal bundle $\nu S^n$ is diffeomorphic to $S^n \times \mathbb{R}$.
Comments: Since the sphere is a submanifold of codimension 1, I believe the answer is true, but I do not know how to prove it
They are diffeo as bundles:
the element $(x, t\cdot x)$ maps to $(x,t)$.
And you can do that for any hypersurface determined by a function with non-vanishing gradient. So indeed the normal bundle is trivial. The tangent one is on the other hand trivial only in some special cases. Note that the sum of these two bundles is a trivial $n$-bundle.
Added:
like @giobrach: stated, a codim $1$ submanifold of an orientable manifold has a trivial normal bundle if and only if the submanifold is orientable.