The {sup 27}Al and {sup 63}Cu spin-lattice relaxation rates were found to contain a large T{sup 2} term over a wide temperature range below 400K in several thermodynamically stable quasicrystalline alloys, including i -AlCuRu, i -AlPdRe, i -AlCuFe, and the crystalline approximant phase {alpha}-AlMnSi. The relaxation mechanism is proven to be electronic in origin. Such nonlinear temperature dependence is shown to be a clear signature of sharp features in the density of states at the Fermi level. The estimated width of this sharp feature is on the order of 20meV. {copyright} {ital 1997} {ital The American Physical Society}