We have used the Australia Telescope National Facility Mopra 22m millimetre telescope to search for emission from the 85.5-GHz and 86.6-GHz transitions of methanol. The search was targeted towards 22 star formation regions which exhibit maser emission in the 107.0-GHz methanol transition, as well as in the 6.6-GHz transition characteristic of class II methanol maser sources. A total of 22 regions were searched at 85.5 GHz resulting in 5 detections, of which 1 appears to be a newly discovered maser. For the 86.6-GHz transition observations were made of 18 regions which yielded 2 detections, but no new maser sources. This search demonstrates that emission from the 85.5- and 86.6-GHz transitions is rare. Detection of maser emission from either of these transitions therefore indicates the presence of special conditions, different from those in the majority of methanol maser sources. We have observed temporal variability in the 86.6-GHz emission towards 345.010+1.792, which along with the very narrow line width, confirms that the emission is a maser in this source. We have combined our current observations with published data for the 6.6-, 12.1-, 85.5-, 86.6-, 107.0-, 108.8- and 156.6-GHz transitions for comparison with the maser model of Sobolev & Deguchi (1994). This has allowed us to estimate the likely ranges of dust temperature, gas density, and methanol column density, both for typical methanol maser sources and for those sources which also show 107.0-GHz emission.