The Australia Telescope National Facility Mopra millimetre telescope has been used to search for 95.1-GHz class I methanol masers towards 62 6.6-GHz class II methanol masers. A total of 26 95.1-GHz masers were detected, 18 of these being new discoveries. Combining the results of this search with observations reported in the literature, a near complete sample of 66 6.6-GHz class II methanol masers has been searched in the 95.1-GHz transition, with detections towards 38 per cent (25 detections; not all of the sources studied in this paper qualify for the complete sample, and some of the sources in the sample were not observed in the present observations).There is no evidence of an anticorrelation between either the velocity range, or peak flux density of the class I and II transitions, contrary to suggestions from previous studies. The majority of class I methanol maser sources have a velocity range that partly overlaps with the class II maser transitions. The presence of a class I methanol maser associated with a class II maser source is not correlated with the presence (or absence) of main-line OH or water masers. Investigations of the properties of the infrared emission associated with the maser sources shows no significant difference between those class II methanol masers with an associated class I maser and those without. This may be consistent with the hypothesis that the objects responsible for driving class I methanol masers are generally not those that produce main-line OH, water or class II methanol masers.