Either, both, or none, depending on the inclinations of the besiegers. Their actions would be determined by several factors.

One, the strength of the work and the garrison defending it. If it was a modern, heavily garrisoned castle, it might be impossible to storm, or the cost in lives and, commensurately, the army's fighting power, might be considered excessive. If it was a weakly garrisoned baronial castle, or a town with a large extent of vulnerable wall, it might be taken by storm.

Two, and this is directly tied to the first, the strategic position the castle occupies. Is it an end in and of itself, or is it merely a stepping stone to one's ultimate goal? How important is it that it be captured quickly? Will a long siege sufficiently use up the campaigning season so that one's army will be compelled to retire even if successful? Or will merely taking the work render the enemy's situation untenable?

Three, what are the prospects for the castle's relief? Is the king or lord to whom it belongs powerful and canny? Can he raise troops quickly and march them decisively to its aid? Or is he cautious and unsure, likely to dither and delay?

Four, what is the castle's logistical situation. Does it rely for fresh water on a stream or small river? If so, this might be dammed. If it depends upon wells, these might run dry after sustained use. Is the castle in proper condition to withstand a siege, with plenty of nonperishable foods in store? Is the garrison very large and likely to eat up these stocks?

A great example of a very long siege encompassing virtually every method of siegecraft is the siege, in the fall and winter of 1203-1204, of Chateau Gaillard by the French king Philip II Augustus. Chateau Gaillard was an English royal castle, constructed under the direct supervision of Richard I, garrisoned very stoutly, with an able castellan, and in a place absolutely vital to the defense of English-controlled Normandy. It occupied a very strong physical position, being located on a high bluff overlooking the Seine, and was approachable only from one direction. The castle consisted of, from innermost to outermost, a very strong keep, an inner wall, an outer, curtain wall, an outer work which guarded the bridge between the castle and the plateau beyond, and a rock-cut ditch as the outermost defense. The following may be useful in conceptualizing: http://upload.wikimedia.org/wikipedia/commons/3/33/Plan.Chateau.Gaillard.png

Philip was initially content to wait the defenders out. He may have been reassured of the wisdom of this course of action by the knowledge that the people of the nearby town of Andely - some 1,700 - had taken refuge within the castle. The castle had been provisioned so that its garrison of 500 might last a year; obviously, a more than 300% increase in the population would shrink that drastically. The castellan, Roger de Lacy, expelled first 500, then 500 more of the townspeople; they were allowed to pass through the lines of besiegers. When he attempted to expel a third group, the French refused them passage, and they turned back, only to find the gates shut to them. They remained in the ditch for some three months, suffering fearfully from hunger and cold, before Philip grudgingly allowed them to be fed and removed.

Philip became impatient in February or early March and ordered an attempt made to undermine a tower of the outer work. The outer work was not built on a stone foundation, like the rest of the castle, and was thus vulnerable to the technique. By undermining, we mean to say that the besiegers dug tunnels, propped up with wooden supports, under a tower of the outer work, filled the tunnel with combustibles, and set fire to them before retreating. This caused the collapse of a tower, and opened a great hole into the outer work. The defenders fell back to the walls proper and prepared themselves, as best they could, for the next stage.

The outer wall was taken by stealth and storm, at least according to Philip's chronicler. Supposedly it was accessed by a small group of French soldiers climbing up a latrine shaft. This is probably apocryphal, and the most likely point of access would seem to be the newly-built chapel, which had not been part of Richard's design and had been shoehorned in by his successor, John. Once upon the walls, they were able to open the gates and let the besieging army in.

The inner walls were taken by means of a breach. A catapult (this is a generic term for a variety of stone-throwing engines) opened a hole in the walls, and the defenders rushed to defend it. They were overwhelmed - by this point, their numbers were down to less than 200 men, and their besiegers numbered some 10,000-12,000 - and captured before they could retreat within the keep.

Thus passed Chateau Gaillard, darling of Richard, and victim of John, who in 7 months had not even managed more than a token effort at relieving it. With it went English Normandy, not to be regained for better than 130 years.