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]
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