What regulatory load cases drive the structural design of aircraft?
The applicable US regulation is FAR 25.301-563. This obviously goes into a lot of detail about the different aspects that should be taken into account. I will try to condense the cases as much as possible, but many are conditional and relate to various other conditions. Many cases include some subjectivity as to what the FAA will require. Every certification is unique.

While the requirements are somewhat broken down into the general areas of the plane, it's impossible to say exactly which areas of the plane will be critically affected by a certain case. The impact depends on the design and configuration of the specific plane.

There are the upper level requirements:


  (a) The structure must be able to support limit loads without detrimental permanent deformation. At any load up to limit loads, the deformation may not interfere with safe operation.
  
  (b) The structure must be able to support ultimate loads without failure for at least 3 seconds...
  
  (e) The airplane must be designed to withstand any vibration and buffeting that might occur in any likely operating condition up to VD/MD, including stall and probable inadvertent excursions beyond the boundaries of the buffet onset envelope.


Definitions of various important values (which will have structural implications):

VA - Design maneuvering speed
VB - Design speed for max gust intensity
VC - Design cruising speed
VD - Design dive speed
VDD - Design drag device speeds
VF - Desgin flap speeds
VMC - Minimum control speed with critical engine inoperative
VS0 - Stall speed with landing flaps
VS1 - Stall speed, flaps retracted
CN - Normal force coefficient

All strength requirements must be met at all places within the maneuvering envelope, particularly A-I on the following V-n diagram:




Max pitch control deflection at VA (25.331)
Specified pitch control displacement (25.331)
1.5g pullup (25.335)
Discrete gust, vertical and lateral (25.341)


VC: varies linearly


+/- 56.0 ft/sec - sea level
+/- 44.0 ft/sec - 15000 feet
+/- 26.0 ft/sec - 50000 feet

VD: 0.5 times VC values

Continuous gust, vertical and lateral (25.341)
Fuel/oil loads from zero to maximum (25.343)


Addtional factors if there is a "structural reserve" of fuel considered

High lift devices (25.345)


If flaps are to be used for takeoff, approach, or landing, limit load factor of 2.0


+/- 25 ft/sec normal to flight path

Separate conditions, limit load factor of 1.0


Propeller slipstream from max continuous power at VF
Propeller slipstream from takeoff power at >= 1.4 times VF and max weight
Head-on gust of 25 ft/sec

If flaps are to be used en route:


Maneuver to positive limit load factor
Discrete vertical gusts

Maneuver load factor of 1.5 g at max takeoff weight, flaps in landing configuration

Roll maneuvering (25.349)


Airplane load factor 0 and 2/3 of positive maneuvering factor


Steady rolling velocities
Maximum angular acceleration
VA Sudden deflection of aileron to the stop
VC Aileron deflection for same roll rate as VA case
VD Aileron deflection for at least 1/3 of VA case

Unsymmetrical gusts

Yaw maneuvering (25.351), speeds from VMC to VD


Sudden max rudder deflection (conditional definition)
At that max deflection:


Yaw to overswing sideslip angle
Yaw to static equilibrium sideslip angle


Rudder suddenly returned to neutral



Engine torque (25.361)


Limit engine torque at takeoff power and prop speed, 75% limit loads of V-n condition A
Limit engine torque at max continuous power and prop speed, 100% limit loads at V-n condition A
For turboprops, previous conditions with prop control system malfunction
For turbine engines:


Torque from sudden engine stoppage
Torque from max engine acceleration


Engine/APU side loads (25.363)


Limit side load factor

Pressurized compartments (25.365)


All flight loads in addition to pressure differential 0 to max relief valve setting
External pressure distribution in flight
Landing loads combine with max pressure allowed at landing
Pressure including safety factor for high altitude aircraft (>= 45,000 feet)
Sudden release of pressure for critical components


Also affects bulkheads, floors, and partitions

Other emergency depressurizations

Engine failure unsymmetrical loads (25.367)


Failure of critical engine (for turboprops, also failure of drag limiting system)


Speeds from VMC to VD, due to fuel flow interruption
Speeds from VMC to VC, due to mechanical failure

Includes pilot corrective action

Gyroscopic loads (25.371)


Must be considered at max appropriate RPM for certain other cases

Speed control devices (25.373)


Symmetrical and yaw maneuvers and gust conditions, each setting with max speed


If automatic/load limiting features, include corresponding speeds/positions


Control sufrace/system loads (25.391)


Considering other limit load cases


Loads parallel to hinge line 
Pilot effort effects
Trim tab effects
Ground gust conditions
Unsymmetrical loads
Auxiliary aero surfaces


Landing load condtions (25.473)


Descent at 10 fps, design landing weight
Descent at 6 fps, design takeoff weight
Both cases should be considered in the following cases:


Level landing
Tail-down landing
One-gear landing
Side loads

Rebound landing
Ground handling
Taxi, takeoff, landing roll
Braked roll
Turning
Tail wheel yaw
Nose wheel yaw/steering
Pivoting
Reversed braking
Towing loads
Unsymmetrical ground loads

Seaplanes (25.521)


Step, bow, stern landings
Unsymmetrical landing
Hull/main float bottom pressures
Auxiliary float loads
Seawing loads

Emergency landing (25.561)


Occupants must have reasonable chance of escaping serious injury in minor crash landing if occupant experiences the following forces:


Upward 3.0g
Forward 9.0g
Sideward 3.0g on airframe, 4.0g on seats
Downward 6.0g
Rearward 1.5g

Large masses should not:


Cause injury to occupants
Penetrate fuel tanks
Block escape

Protection of occupants in dynamic conditions
Ditching strength

Fatigue and damage tolerance (25.571)


Residual strength with certain failed structure:


Limit symmetrical maneuvers
Limit gust conditions
Limit rolling conditions
Limit yaw conditions
Pressure differentials
Limit ground loads

Safe-life evaluation, if failure conditions are not practical
Sonic fatigue
Safe flight with damage from:


4 pound bird at VC at sea level and 0.85VC at 8,000 feet
Uncontained fan blade impact
Uncontained engine failure
Uncontained high-energy rotating machinery failure