I can give you an idea of what a crusader army was doing and you may be surprised by how much activity was necessary to keep a besieging army in the field, even when it was static.

Food and water: Although we might think of those trapped within a city as being desperate for food, it was often the other way round in reality. A good-sized Medieval army was unlikely to take a settlement by surprise; during certain stretches of the march through Palestine the Europeans were managing well under 10 miles per day. This meant, when you came before the walls, your enemy was inside with food and you were outside with precious little.

The force which laid siege to Jerusalem in 1099, for example, probably contained at least 16,000 men, not counting the myriad other followers, pilgrims, churchmen, attendants etc. The arid surroundings, often hostile local populations and stifling heat of summer made provisioning the host difficult.

Which brings us to our first answer: soldiers would be actively foraging for food sources and establishing supplies of fresh water. Foraging parties would seek out whatever sustenance they could for the besieging army, which could be slim-pickings if your foe had been expecting you. It was also dangerous; in September 1191 a foraging expedition of the Third Crusade ran into a detachment of Saladin's army in an orchard. Richard I of England rode out to their aid and was only just able to extricate himself to safety.

Likewise, water supplies could not be relied upon with great certainty while in the field. The siege of Jerusalem required 'elaborate schemes of water-carrying over large distances' and any such resources had to be maintained and also defended (Tyerman, p.155). To just keep a large host fed and watered in hostile territory will therefore have occupied a great deal of the time of many in a Medieval army.

Combat: It may seem obvious but a considerable deal of fighting could be expected during sieges, regardless of whether a storm or assault was in progress. Settlements with requisite supplies and manpower would often launch raids or 'forays' out of their fortifications to inflict surprise injuries against the besieging force. For this reason, even an army at rest outside the walls of a city frequently required active defense and patrol.

Take for example the siege of Antioch in 1097-98. At no point was the entirety of the city blockaded by the crusaders because of the scale of the fortifications, which meant soldiers had to be constantly vigilant to movements in and out of the city. Defenders could ambush their attackers or assault them with missiles with a considerable degree of freedom. The somewhat traditional view of static and unchanging battle lines in siege warfare is quite often inaccurate.

Sometimes the state of affairs was more fractious still. If we consider the siege of Acre in 1189-91, during which a Muslim garrison was besieged by Guy de Lusignan who himself in turn was besieged by Saladin, soldiers will have been spending a great deal of time creating and manning trenches, counter-trenches and other fortifications. No-man's land and territorial demarcations shifted. The experience here for the fighting men was less that of inactivity and more one of 'frequent raids and close-combat skirmishes' (Tyerman, p.411).

Siege engines and ladders: Of course a well-provisioned and defended settlement has the luxury of time for inactivity, if they so choose, because it is the besieging force's duty to ultimately take by assault what cannot be taken by blockade. To this end, Medieval soldiers at siege would spend time building, maintaining and deploying siege engines.

It obviously follows that the materials for these weapons, such as wood, also needed to be foraged for, as with food and water sources mentioned above. This was a serous logistical undertaking in terms of manpower, with the Frankish army 'ferrying in timber by the camel-load' as the First Crusade reached its climax (Asbridge, p.95).

Godfrey de Bouillon's siege tower had to be assembled from smaller, collapsible pieces to aid in its mobility. The soldiers were also engaged in building a huge iron-clad battering ram with which to assault the outer fortifications, under supervision from Genoese craftsmen. Together with catapults and ladders, this undertaking represented 'a furious programme of construction' over three weeks (Asbridge, p.95).

Sex

This is not my particular area, certainly others will know far more about women in siege camps, but considerable numbers of prostitutes would gather wherever a field army assembled, to engage in their business. This was acknowledged (and often loathed) by chroniclers: Gillingham quotes Ambroise on the Third Crusade army, 'Back to the host the women came, To ply their trade of lust and shame'. We may say that from antiquity to the modern day that death and sex are often found in the same places, being the oldest professions as they are.

Hopefully this was a useful flavour of how much activity could be observed in a besieging army, as well as the sheer logistical efforts needed to keep it functioning. Other people might be able to impart additional information on things like religious services in camp, or perhaps leisure/gambling? These were some of the more pragmatic realities of sieges from campaigns I'm familiar with.

Read more:

C. Tyerman, God's War (2007)

I. Gillingham, Richard I (2002)

T. Asbridge, The Crusades (2012)

R. C. Smail, Crusading Warfare (1995)

Edit: Formatting.