The following publication by Gottlieb et al. (linked here) offers an excellent collection of 1H and 13C chemical shifts to commonly-used NMR solvents.
From the 1H NMR spectrum below, the first step is to differentiate the signals pertaining to the unknown from those of the residual solvent. The key here lies with the integral values.
The integrals for the multiplets at 2.4, 2.7 and 3.2 ppm are very close to whole numbers and as such point to one compound (labeled A on the spectrum). The integrals for the multiplets at 1.2 and 3.5 ppm indicate a second compound (labeled B on the spectrum).
Diethyl ether exhibits a typical 1H NMR pattern, a triplet at ~1.2 ppm (CH3) and a quartet at ~3.5 ppm (CH2) in CDCl3 at room temperature. Ethanol exhibits an identical coupling pattern to diethyl ether but the chemicals shifts are slightly higher at ~1.7 (CH3) and 3.7 ppm (CH2). Based on the similarities between the chemical shifts, the residual solvent appears to be diethyl ether.
A special thanks goes to Bogdan, Carlos, R. Murugan and Djalma for their comments.