AFAIK there was only one quite rare model of the Thompson (the 1923 model), to which a bayonet could be fitted to and that was the American style one in any case. Otherwise one could not be fitted.
Jeff's pretty much on the money for me as regards what was carried. Regimental 'orders of the day' covered everything that might or might not be worn (on any particular day/week/month) and the relative strictness of your sergeant in prosecuting those orders being the final word, barring those of the CSM or RSM, which were largely four letter ones in any case.
'Officially' even Bren gunners were issued rifles which they were 'supposed' to carry along with the Bren itself, but you never see them actually doing that in photos... likewise you do not see Bren Guns on ceremonial duty photos. 2" Mortars and Boys AT rifles were held as platoon weapons, so their chosen operators, when they were required, were also rifle-armed men.
The bayonet was a really useful tool though, as a few of you have mentioned, rumour has it that you could actually use it against the enemy too if you had to. I suspect that quite a few people who did not really need one probably carried one and those who did not have one (i.e. those 'officially' issued pistols) probably acquired one. Given the numbers I and a few here have come across over the years, it sounds like it must have been the most commonly 'lost item' reported on demob in 1945... barring the clasp knife.
For me then, in a purely general, but by no means universal sense, it's bayonets for all, unless you have pips on your epaulettes, or were wearing just the belt and pistol pouch, to which adding a bayonet might mean you were 'out of uniform'... one of the worst crimes known to man, other than losing something of course.
