Nothing scientific - just anecdotal but I've had quite a few conversations over the years with folk who don't cycle but do travel in motor vehicles - either as drivers or passengers - and the number of times someone has believed that cycling on some dual carriageways is not permitted (where it is) is astounding. It seems that [false] assumptions of the lack of rights of cyclists (and other non motorists) and the rights of motorists on "fast roads" are becoming more entrenched. When I was younger, I often used "fast" dual carriageways to cover long distances on bike and whilst feeling less safe than on some other roads, did not feel especially fearful; on those same roads today, I would be very scared.
Of course, there's a lot more motor traffic on the roads nowadays and that must play a part in it, but I think there's more to it than that, and that the false assumptions mentioned above are, to an extent, playing a dual role of both cause and effect (people think cyclists ain't allowed so they drive accordingly, and because people drive like that, it is assumed that cyclists ain't allowed). I also recall when younger that HGV drivers were usually less scary than car drivers, driving to a higher and safer standard - but I don't notice that nowadays.
Dual carriageways near me that scare me silly (I've never cycled on them - it's just the thought of it) are the A45 between Coventry and Birmingham, and the A46 Kenilworth/Warwick bypass.