Similiar to how current Special Forces are, such as rapid/surprise assault and elite soldiers
Firstly, I can't speak for Rome, but I can say something about the military structure of France and England in the High Middle Ages. Basically, regular, standing forces were not a thing during this period. Armies were raised for a specific campaign. Mercenaries were hired, levies called up, and disbanded following its conclusion. Rulers simply didn't have the resources to maintain a professional army.
But clearly, forming an army entirely from scratch was problematic then, as it is now, and too slow to react to emergencies. The solution to this was the maintenance of household men, what are sometimes called retinues. Virtually every noble would have landless knights attached to his household at all times, and the wealthier and more powerful had correspondingly more men. We have the most information on the households of the 13th century English kings, John and Edward I. Edward, at times, had around 600 knights attached to his household, though they were not all with them at the same time. These men could be temporarily granted royal lands to support themselves, with the understanding that they would not hold them perpetually, and they would respond immediately to summons in a way that the nobility could not be counted on to. In event of war, they formed the nucleus of the army, performing the functions equivalent to staff work, as well as serving as bodyguards.
That is not how 'special forces' are defined today. These definitions (and your own) are utterly anachronistic when applied to the Middle Ages in Europe. 'Elite' soldiers tended to be a product of individual capability and experience rather than of specific training as modern 'elite' soldiers are.
Even Yuval Harari's book which attempts to apply another modern anachronism 'special operations' notes that there was no such this as a medieval 'special force'. When extraordinary endeavours needed to be undertaken normal soldiers might do it, or those who had a reputation for being good at something might do it. There is no real indication that anyone trained for these types of activities.
Harari's book has interesting tidbits but no analytical framework or cohesive structure to back it up. I actively discourage anyone reading it without a proper understanding of the moral and practical contexts of warfare in the Central to Late Middle Ages: Special Operations in the Age of Chivarly, 1100-1550, Oxford, 2007.
A refinement of this question that might lead to more fruitful discussion might be to ask if and how medieval forces carried out the type of operations that fall under the umbrella of modern-day special operations forces, most of which have little to do with the door-kicking they're best known for.