I am unable to visualize any case where angular momentum and angular velocity of an object are not parallel.
[Physics] Example where angular momentum and angular velocity are not parallel
angular momentumangular velocityhomework-and-exercisesnewtonian-mechanicsrotational-kinematics
Related Solutions
In general, the angular momentum for a rigid body is $\vec{L}=I\vec{\omega}$.
For the special case of a point particle $\vec{r}$ from the axis of rotation, we have $I=mr^2$ and $\vec{\omega}=\frac{\hat{r}\times\vec{v}}{r}$, or $\omega=\frac{v}{r}\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{r}$.
In this case, the angular momentum becomes $L=mrv\sin\theta$.
Consider a thin rectangular block with width $w$, height $h$ resting along the xy plane as shown below.
The mass of the block is $m$. The mass moment of inertia (tensor) of the block about point A is
$$ {\bf I}_A = m \begin{vmatrix} \frac{h^2}{3} & -\frac{w h}{4} & 0 \\ -\frac{w h}{4} & \frac{w^2}{3} & 0 \\ 0 & 0 & \frac{w^2+h^2}{3} \end{vmatrix} $$
This was derived from the definition (as seen on https://physics.stackexchange.com/a/244969/392)
If this block is rotating along the x axis with a rotational velocity $$ \boldsymbol{\omega} = \begin{pmatrix} \Omega \\ 0 \\ 0 \end{pmatrix} $$ then the angular momentum about point A is
$${\bf L}_A = m \Omega\,\begin{pmatrix} \frac{h^2}{3} \\ -\frac{w h}{4} \\ 0 \end{pmatrix} $$
As you can see, there is a component of angular momentum in the y direction. The angular momentum vector forms an angle $\psi = -\tan^{-1} \left( \frac{3 w}{4 h} \right)$
In the figure below you see the direction of angular momentum, and the circle about which the center of mass is going to orbit due to precession.
Best Answer
In the basic discussion of angular momentum where something is rotating around a fixed symmetrical axis
$\vec{L}=\vec{r}\times\vec{p}$
reduces to
$\vec{L}=I*\vec{\omega}$
Like in this animation where each vector is colored appropriately:
However angular velocity and angular momentum can have different directions in two cases: If the axis of rotation is not symmetric or the axis of rotation is moving.
Here's an example:
You can see that $\vec{L}=\vec{r}\times\vec{p}$ is not the same direction as $\vec{\omega}$ nor would the simplification $\vec{L}=I*\vec{\omega}$ be correct.
The position vector $\vec{r}$ is the vector between the reference point and the mass (note these problems are ignoring the mass of the rod), only in simple rotational cases like the first case is it perpendicular to $\vec{\omega}$. In a system of masses for example these vectors to the masses about a reference point can be complex. It is much easier to take the reference point as the center of mass. In each case $\vec{r}$ is the positional vector between your reference point and the mass and their composite angular momentums will superimpose (add) together.