Annie's Molten Shield
Blitz's Mana Shield
Janna's Eye of the Storm
Morgana's Black Shield
Rammus' Defense Curl
Shen's Stand United & Feint
Sion's Death Caress
Sivir's Spell Shield
Udyr's Turtle Stance
Clearly no one can answer ur question as no one could've possibly tested out all those champs on the same team.
However speculation can be done;
1. Abilities that mitigate damage (Rather than block or absorb) go first.
I.e. rammus' defense curl
2. Shields that are personal to the character (i.e. Shen's Feint, Sion's Death Caress, Udyr's Turtle Stance, Mord's Passive, Blitz's Mana Shield) always get hit last.
3. Shields that are non-personal (i.e. Kayle's Intervention, Shen's Stand United, Morg's Spell Shield, Janna's Eye of the Storm) always get hit first.
4. Shields that proc effects other than just blocking or mitigating it (regardless of point 2 and 3) always go first (spell shield, annie's molten shield)\
Now the order of point 3 is hard to say.
I'd say the order of effect (as in the shield that takes hit first) would be;
1. Morg's Black Shield (in combination) Kayle's Ult
One blocks spells and spell damage and the other physical. Together, the person shielded shouldn't die.
2. Shen's Stand United
3. Janna's Eye of the Storm
Again its pure speculation. I guess the order I provided for "point 3" can be tested.