The first case of spectral stability looked at was the explicit Euler. Spectral Stability is defined as stability which depends on the eigenvalues.
$$\text{lim}_{k\rightarrow \infty} || \mathbf{V}_k || = 0$$
It is worth taking a worst case look at stable explicit Euler again. That is
$$\mu = \dfrac{1}{2}$$
which is the maximum $\mu$ value permissible for stability.
$$\mu = \dfrac{\Delta t}{\Delta x^2}$$
which means that small $\Delta x^2$ would require tiny $\Delta t$ while preseving the $\mu$ criterion. E.g
$$\Delta x = 10^{-3} \Rightarrow \Delta t = \dfrac{1}{2} \times 10^{-6}$$
This means that stability requires lots of time-steps! Therefore explicit Euler is inefficient and a bad method.