Example of a nontrivial subbundle of a trivial vector bundle

You're right; a good example is the Möbius bundle on $\mathbb{S}^1$ sitting inside the trivial bundle $\mathbb{S}^1\times\mathbb{R}^2$.

In the image below, you can think of $\mathbb{S}^1$ as being the central circle of the whole apparatus, and the "square" cross-section of each slice as being $\mathbb{R}^2$, which is after all diffeomorphic to $(0,1)^2$.

After some quick modifications to my code from this question, here is a depiction:

          

For my own future reference / anyone else, the edited parts of the code are

NewFaces[R_,r_,s_,t_] := {F[R][t,s,r], F[R][t,r,s], F[R][t,-r,-s], F[R][t,-s,-r]}

NewEdges[R_,r_,t_] := {F[R][t,r,r], F[R][t,-r,r], F[R][t,r,-r], F[R][t,-r,-r]}

ThickMobius[R_,r_,u_] := Show[ParametricPlot3D[NewFaces[R, r, s, t], {s, -r, r},
{t, 0, 2 Pi}, PlotStyle -> {Blue, Opacity -> 0.15}, PlotPoints -> {2, 50}, 
Mesh -> None, Boxed -> False, Axes -> None], ParametricPlot3D[Strip[R, r, s, t],
{s, -r, r}, {t, 0, 2 Pi}, Mesh -> None, PlotStyle -> Red, PlotPoints -> 50], 
ParametricPlot3D[NewEdges[R, r, t], {t, 0, 2 Pi}, PlotStyle -> {Darker[Blue], 
Thickness[u]}, PlotPoints -> 30]]

Take any compact manifold $M$ embedded in $\mathbb R^N$, with non-trivial tangent bundle (like $S^2\subset\mathbb R^3$). Then $TM$ sits inside the trivial vector bundle $M\times \mathbb R^N$.

As any compact manifold can be embedded into a suitable $\mathbb R^N$, this means that any tangent bundle of a compact manifold is subbundle of a trivial one. This is even more generally true for any finite-dimensional vector bundle over a compact manifold instead of the tangent bundle.