To put it very simplistically, from mostly a practical view: abs: cannot be differentiated around 0; has multiple minima; the error space has sharp ridges power

To put it very simplistically, from mostly a practical view: abs: cannot be differentiated around 0; has multiple minima; the error space has sharp ridges power 4: way too sensitive to noise power 3: var(x+y) != var(x) + var(y)