This paper presents the third in a series of single-dish studies of molecular abundances in the envelopes around a large sample of 18 low-mass pre- and protostellar objects . It focuses on typical grain mantle products and organic molecules , including H _ { 2 } CO , CH _ { 3 } OH and CH _ { 3 } CN . With a few exceptions , all H _ { 2 } CO lines can be fit by constant abundances of 7 \times 10 ^ { -11 } –8 \times 10 ^ { -9 } throughout the envelopes if ortho- and para lines are considered independently . The current observational dataset does not require a large H _ { 2 } CO abundance enhancement in the inner warm regions , but this can also not be ruled out . Through comparison of the H _ { 2 } CO abundances of the entire sample , the H _ { 2 } CO ortho-para ratio is constrained to be 1.6 \pm 0.3 consistent with thermalization on grains at temperatures of 10–15 K. The H _ { 2 } CO abundances can be related to the empirical chemical network established on the basis of our previously reported survey of other species and is found to be closely correlated with that of the nitrogen-bearing molecules . These correlations reflect the freeze-out of molecules at low temperatures and high densities , with the constant H _ { 2 } CO abundance being a measure of the size of the freeze-out zone . An improved fit to the data is obtained with a ‘ drop ’ abundance structure in which the abundance is typically a few \times 10 ^ { -10 } when the temperature is lower than the evaporation temperature and the density high enough so that the timescale for depletion is less than the lifetime of the core . The location of the freeze-out zone is constrained from CO observations . Outside the freeze-out zone , the H _ { 2 } CO abundance is typically a few \times 10 ^ { -9 } -10 ^ { -8 } . The observations show that the CH _ { 3 } OH lines are significantly broader than the H _ { 2 } CO lines , indicating that they probe kinematically distinct regions . CH _ { 3 } OH is moreover only detected toward a handful of sources and CH _ { 3 } CN toward only one , NGC 1333-IRAS2 . For NGC 1333-IRAS2 , CH _ { 3 } OH and CH _ { 3 } CN abundance enhancements of two-three orders of magnitude at temperatures higher than 90 K are derived . In contrast , the NGC 1333-IRAS4A and IRAS4B CH _ { 3 } OH data are fitted with a constant abundance and an abundance enhancement at a lower temperature of 30 K , respectively . This is consistent with a scenario where CH _ { 3 } OH probes the action of compact outflows on the envelopes , which is further supported by comparison to high frequency , high excitation CS J = 10–9 and HDO line profiles which uniquely probe warm , dense gas . The extent to which the outflow dominates the abundance enhancements compared with the passively heated inner envelope depends on the filling factors of the two components in the observing beam .