It's my understanding that siege warfare was less about breaching a single point in a fort's/castle's/city's defenses, and more about wearing the besieged down to the point where they either attacked you, or gave up. Siege engines were thus less of a direct offensive tool, and more of a harassment and/or weakening device. They were used to soften up the larger object's defenses, and slowly take a toll (physically and mentally) on the soldiers manning that city, fort, etc. There were obvious exceptions to this; Cromwell, for instance, did try to focus his fire on a two specific breach points during the Siege of Basing House. During the American Civil War, there was the disastrous Battle of the Crater in front of Petersburg, which didn't involve siege machines, per se, but did represent an attempt to end a siege via a direct, focused attack on a single point.

Caesar engaged in maybe the most famous and heavily studied siege in history at Alesia. To this day, Caesar's method of circumvallation (digging in around your target with the purpose of preventing their escape out), and contravallation (establishing a second perimeter behind your siege line to prevent reinforcements from coming in) is still studied. THIS is how you run a successful siege. The reason this was so effective was that it precluded the need for an offensive army to risk combat casualties by attempting an assault on a well-fortified position, for a well-run siege should eventually wear a defender out from a lack of supplies.

[Sources - Samuel Gardiner, 'History of the Great Civil War'; Bruce Catton, 'A Stillness at Appomattox'; Adrian Goldsworthy, 'Roman Warfare']

I can't really speak on medieval siege battles, but during the Early Modern period, a standardized, scientific method of taking star-shaped artillery fortresses emerged, most famously standardized by Vauban.

First, the besieging army would construct trenches facing in towards the fortification, and outwards, in case they needed to repel an attacking army. From there, they would dig trenches forward (a process called sapping, with zig-zagging forward trenches called saps) towards the spot in the enemy fortifications judged to be the weakest. After getting relatively close to the bastion they sought to break, the army would dig a trench parallel to the general direction of the wall. From there, they would lay suppressive fire on the cannons and troops firing from the fortress. They would then sap forward again to construct a second parallel, to really start damaging the wall they were trying to break down. Repeating the sapping process one more time, they would construct a third parallel trench close to the trench surrounding the enemy fortress, called the Covered Way. The Covered Way served as a shelter and jumping off point for the fortress garrison as it attempted to raid the enemy's siege camp, and capturing it was the key moment of a siege. Most forts surrendered shortly afterwards, but if not, the walls could be breached at close range by concentrated heavy siege cannon.

Obviously, this is a lot of work that goes into breaching modern fortifications, so attackers generally concentrated their efforts on the sections of works that would be most swiftly breached.

It depended more on the fortifications themselves and the availability of siege engines than anything else.

If we look at Mehmet II's siege of Constantinople in 1453, he pretty much just shot the hell out of the wall everywhere. The Theodosian Walls, though outmoded by that point, didn't really have any specific weaknesses (except where the wall abutted Blachernae, but the Ottomans didn't have much luck forcing that point). So Mehmet pretty much just shot everything within reach of his batteries. He focused fire on where breaches had occurred (this was to kill defenders who were erecting gabions more than to simply expand the gaps, though it had that benefit), and would direct fire on specific sites to either force a breach (Blachernae, unsuccessfully) or on specific locations prior to an assault, and at one point he seems to have directed artillery fire on a gate because he could, but it's not like there was a single thing that the guns focused on during the entire siege.

Siege weapons that rely on indirect fire (trebuchets, howitzers) were also rather less accurate than direct fire weapons (springald, guns). The trade-off here is that direct fire weapons, especially prior to the widespread use of guns, were significantly less powerful. If you're trying to use a springald to knock down a wall, you're doing something wrong. Still, a weapon fired on a parabolic arc is going to have a harder time hitting a structure that's rather tall and narrow than a weapon that fires on a more or less flat trajectory.

Mehmet II of the Ottoman Empire sieged cities and castles by surrounding the entirety of it, capturing any other nearby forts, and then focusing artillery on a weak point in order to force a breach. This was the norm before star forts replaced more traditional castles as the most effective means of defensively multiplying limited manpower - ie the strength of the garrison. Typically, once it looked like a breach would be made, the defenders would surrender.

I don't think there's a clear answer to your question, because tactics changed based on the fortification being assaulted, the available weaponry and supplies, and the culture of the attackers.

There are a few examples from the First Crusade that demonstrate this. In 1099 they assaulted the walls of Jerusalem using two siege towers at different points of the wall to spread out the defenders. They had to assault because they didn't have the supplies for an actual siege. However, at Antioch the Crusaders believed that the fortifications were too strong to be taken in direct assault, so they attempted to starve the city, besieging it for 9 months before taking it via treachery.