  Complex organic molecules (COMs) have been detected in a few Class 0
protostars but their origin is not well understood. Going beyond studies of
individual objects, we want to investigate the origin of COMs in young
protostars on a statistical basis. We use the CALYPSO survey performed with the
IRAM PdBI to search for COMs at high angular resolution in a sample of 26
solar-type protostars, including 22 Class 0 and four Class I objects. Methanol
is detected in 12 sources and tentatively in one source, which represents half
of the sample. Eight sources (30%) have detections of at least three COMs. We
find a strong chemical differentiation in multiple systems with five systems
having one component with at least three COMs detected but the other component
devoid of COM emission. The internal luminosity is found to be the source
parameter impacting the most the COM chemical composition of the sources, while
there is no obvious correlation between the detection of COM emission and that
of a disk-like structure. A canonical hot-corino origin may explain the COM
emission in four sources, an accretion-shock origin in two or possibly three
sources, and an outflow origin in three sources. The CALYPSO sources with COM
detections can be classified into three groups on the basis of the abundances
of oxygen-bearing molecules, cyanides, and CHO-bearing molecules. These
chemical groups correlate neither with the COM origin scenarii, nor with the
evolutionary status of the sources if we take the ratio of envelope mass to
internal luminosity as an evolutionary tracer. We find strong correlations
between molecules that are a priori not related chemically (for instance
methanol and methyl cyanide), implying that the existence of a correlation does
not imply a chemical link. [abridged]
