Signal Reconstruction in Compressed Sensing with a Simple Vector Signal as an Example
Better is hard to measure unless there is a well defined measure.

Yet in the classic Compressed Sensing model (Sparse Representation) the Holly Grail is actually the $ {L}_{0} $ Pseudo Norm.

The problem is $ {L}_{0} $ isn't really a norm and hence the optimization problem isn't convex.
Moreover, it is not smooth (It has discrete properties) and the problem:

$$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} \quad \text{s.t.} \quad {\left\| x \right\|}_{0} \leq k $$

Is actually NP hard. Namely we don't know how to solve it efficiently.

We can prove that under some cases solving the following problem:

$$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} \quad \text{s.t.} \quad {\left\| x \right\|}_{1} \leq k $$

Yields the same solution. Yet this form is Convex and we have many effective tools to solve it.

Hence it is preferable to use this model in the cases it is good enough.