I am particularly interested in the types of rifles that troops serving in Africa would use, especially during the 1890's. Did they have any kind of repeating rifles, or were they still using single shot rifles? Any specific details would be appreciated.
They would have started the decade with Lee-Metford bolt-action 8-10 shot rifles and ended the decade with Lee-Enfield bolt-action 10 shot clip rifles. There were probably still some Martini-Henry level-action single shot rifles as well as their updated version being used at the time but the 1890s seems to be when the single shot rifle was phased out in favor of those with magazines.
From "The Modern Traveller", published 1898:
I never shall forget the way
He marked them in their rude advance,
More seriously, Wikipedia provides this account of the Battle of Omdurman, 2 September 1898
Casualties are listed as:
The Maxim guns were single-barrelled of various styles firing 600 rounds/minute.