To say that a British regular soldier from 1700-1856 would be proficient with a sword is rather untrue as the primary weapon of the British "redcoat" was a musket (or rifle) plus a bayonet but not a sword. Until after the Seven Years War they also carried a short sword but this was rarely used. Officers did carry swords but until the Napoleonic wars they also carried spears called spootoons (though they often exchanged these in favour of a musket). On average they would have been proficent with their swords.

And if you are referring to Homeric style duels where two men run out and engage in single combat on the field of battle then the answer is "sort of". There are many stories for example when a cavalry man would ride forward to encourage his comrades by engaging the opposite enemy cavalry in a duel such as this case at the battle of Friedland in 1807 when the French carabiniers balked at charging the Russian cavalry.

"The Russians waited on the far side of a ditch. The carabiniers balked until an officer jumped the ditch and engaged a Russian officer in a sword duel. His example electrified his troopers and they drove their mounts over the ditch and defeated the Russian heavy horse. Now the carabiniers were unsupported and the Russian cuirassiers reserve regained the advantage."

At Friedland again there was this action when a Russian general charged forward with a regiment of dragoons

"In a display worthy of his ancestors, he galloped ahead of his regiment to engage a French officer in a duel. His cuirassier foe relied upon his armor and turned his head to absorb the blow while thrusting with his long, straight sword. The cuirassier's armor deflected Pahlen's blow while his sword penetrated Pahlen's body. Weakened and in pain, Pahlen feebly tried to turn his horse. The cuirassier thrust again, pinning Pahlen's body to his horse. Their chef's mortal wound enraged the Russian dragoons. They launched a furious charge that drove Nansouty back into the forest southwest of Heinrichsdorf."

It was also not unusual for the forward sentries to engage in individual fights with their counterparts but sometimes their enemy was not as keen for the fight such as this instance that Edward Costello of the 95th Rifles recalled

"One of their videttes, [sentries] after being posted facing English dragoon, of the 14th or 16th [Light Dragoon Regiment] displayed an instance of individual gallantry, in which the French, to do them justice, were seldom wanting.Waving his long straight sword, the Frenchman rode within 60 yards of our dragoon, and challenged him to single combat. We immediately expected to see our cavalry man engage his opponent, sword in hand. Instead of this, however, he unslung his carbine and fired at the Frenchman, who not a whit dismayed, shouted out so that every one could hear him, Venez avec la sabre: je suis pret pour Napoleon et la belle France. Having vainly endeavoured to induce the Englishman to a personal conflict, and after having endured two or three shots from his carbine, the Frenchman rode proudly back to his ground, cheered even by our own men. We were much amused by his gallantry, while we hissed our own dragoon ... "

This could only happen with the cavalry however. They were the ones who had more free reign on the battlefield and could also engage in these actions in recon missions and the like. Infantry would never had the same opportunity to engage in actions such as these. Cavalry troopers were also certainly more likely to have La gloire in their bones.

So all in all somewhat common amongst the cavalry but completly alien amongst the infantry unless anyone can find me a example.