Context. Despite over 50 years of research, many open questions remain about the origin and nature of gamma-ray bursts (GRBs). Linear polarization measurements of the prompt emission of these extreme phenomena have long been thought to be key to answering a range of these questions. The POLAR detector was designed to produce the first set of detailed and reliable linear polarization measurements in the 50 − 500 keV energy range. During late 2016 and early 2017, POLAR detected a total of 55 GRBs. The analysis results of 5 of these GRBs have been reported, and were found to be consistent with a low or unpolarized flux. However, previous reports by other collaborations found high levels of linear polarization, including some as high as 90%. Aims. We study the linear polarization for the 14 GRBs observed by POLAR for which statistically robust inferences are possible. Additionally, time-resolved polarization studies are performed on GRBs with sufficient apparent flux. Methods. A publicly available polarization analysis tool, developed within the Multi-Mission Maximum Likelihood framework (3ML), was used to produce statistically robust results. The method allows spectral and polarimetric data from POLAR to be combined with spectral data from the Fermi Gamma-ray Burst Monitor (Fermi-GBM) and the Neil Gehrels Swift Observatory (hereafter Swift) to jointly model the spectral and polarimetric parameters. Results. The time-integrated analysis finds all results to be compatible with low or zero polarization with the caveat that, when timeresolved analysis is possible within individual pulses, we observe moderate linear polarization with a rapidly changing polarization angle. Therefore, time-integrated polarization results, while pointing to lower polarization, are potentially an artifact of summing over the changing polarization signal and thus washing out the true moderate polarization. We therefore caution against overinterpretation of any time-integrated results inferred herein and encourage the community to wait for more detailed polarization measurements from forthcoming missions such as POLAR-2 and LEAP.