In the entrance room, the display case by the stairs holds a copy of Mysticism and an empty grand soul gem under a very hard lock. A bench sits next to the display case, with a clutter sack by the bench's leg. At the foot of the stairs is another shelf with a clutter sack under it, and at the top of the stairs is a clutter chest and another clutter sack by the door to the upper level. Across the room is a cupboard that also contains clutter with a novice alembic on it. To the left of the cupboard is a corner where two barrels and a sack, all with clutter inside, sit across from the door to the basement.
To the right side of the cupboard, a wooden door opens to a small room. Two beds in the room are commonly slept in by Bothiel, and either Irlav Joral or Caranya. A potted nirnroot lies on a shelf in front of the wooden door, and the sack lying beside it holds more clutter. A cupboard containing clothes is found in the west end of the small room. A clutter chest and a barrel that holds random rare ingredients can be found near the cupboard behind one of the beds.
Mage Quarters Upper Level
Reached by the stairs from the main floor, the upper level hosts many beds which mage apprentices and some other prominent university people can be found sleeping in at night. Six beds are available for those seeking rest, although most of the beds will be filled after 6pm. A large bookcase dominates the northeast wall in the room, and contains many items including a novice alembic and a copy of Dwemer History and Culture. A chandelier hangs quite close to the bookcase, a little too close to the ground. A small table near the door holds a bowl with nine strawberries.
Two sacks sit by the bookcase, near the door. Two more can be found in the north corner beside a bed. Yet another sack sits next to a chest in the west corner of the room. Clutter can be found in each of these containers. There are three drawers in all: one in between two beds in the southern corner has a bottle of Surilie Brothers Wine on it, another with a cheese wedge on it lies at the foot of a bed in the western corner, and the last sits near the northern corner. Each of the drawers contains random articles of clothing. A desk sits in the middle of the room. Another desk lies at the foot of a bed in the southern corner, with a novice mortar & pestle sitting on it. Both desks hold more clutter.
Mage Quarters Basement
The Mage Scholars frequently occupy the basement's beds, and is where Renald Viernis usually rests for the night. Going down the steps, a sack, barrel, and chest can all be found against the wall, each containing clutter. Further down, a sack sits on the bottom step with a desk containing clutter holding a novice mortar & pestle nearby.
The basement opens up to four beds against the basement walls. A bottle of Surilie Brothers Wine sits on drawers that hold clothing near the bed in the middle of the southeast wall. Two sacks sit on the other side of that same bed, one carrying a random grain. A chest containing clutter sits obscured behind the bed in the western corner. In the southern corner, two clutter barrels and a desk can be found, with the desk containing the same items as the other one by the steps.
Once again, the sacks are the only containers that do not respawn in the whole of the Mage Quarters.
Elder Scrolls Wikia entry on this location can be found here.