Sort by when the noise appears
The repair pages describe noises by their trigger: speed, steering and engine RPM. Humming that rises with speed is bearings (shaking is in the BMW Vibration and Steering Shimmy: Brakes, Mounts and Driveshaft), whining when turning is power steering, and a whine that rises with RPM at the front of the engine is a timing chain symptom on the N20. These are reported patterns, not a diagnosis.
Wheel bearings
The front bearing page lists humming or growling that rises with speed, a noise that changes when turning and an ABS or DSC warning in some cases. The hub bolts are single-use and should be replaced, and xDrive cars cost more because the axle has to come apart. See Front Wheel Bearing / Hub.
The rear bearing page lists the same hum plus play when rocking the rear wheel. Rear bearings are pressed in, seized half shafts are common, and one DIYer gave up after two days and paid the indy labor. With the BMW puller tool no alignment is needed. See Rear Wheel Bearing.
Power steering
On E90, E91 and E92 cars with hydraulic power steering, whining when turning, fluid on the reservoir, hoses or under the car, and heavy steering are the signs. If the pump grinds, check the plastic pulley, and leaks are often at the cooler-line O-rings or crush washers, which is a cheap fix. See E90 Power Steering Hoses & Pump. The F30 uses an electric rack instead, where the symptoms are a sudden loss of assist and a steering warning (Electric Steering Rack (F30)).
Differentials and transfer case
Whine from the rear is on the rear differential fluid page, along with preventive service, and on the X3 axle seal page a whirring from the rear at speed if fluid gets low (Rear Differential Fluid Change, G01 X3 Rear Differential Axle Seals). Noise from the front axle area is on the front differential page (Front Differential Fluid (xDrive)), and whining or grinding near the center of the car is on the transfer case actuator page (Transfer Case Actuator (xDrive VTG)).
Belt, A/C and engine front
A squeal or rattle from the belt area with warm A/C and a broken pulley is on the E9x compressor page (A/C Compressor (E9x)), and belt chirps are on the belt kit page (Serpentine Belt, Tensioner & Idler). A whine from the front of the engine that rises with RPM on an N20 is a timing chain symptom (N20 Timing Chain & Guides); for rattles, see the BMW Cold Start Rattle, Whine and Squeal: Causes by Engine.