What are the 10 most important MOSs in the Army?

With these ten MOS''s, your Army could not function. Supply (Logistics) can not operate without Army Contracting. Every MOS is important to mission success, or it would not exist.

You are, SGT Wright, I believe operating under the mistaken assumption that there ARE MOS more important than others. Wrong Answer.
There are Combat Arms, Combat Support, and Combat Service Support MOS. ALL are important. Without the Infantry, Armor, Artillery, Engineers, ADA and Special Ops, there would be no Army, for at it's heart, every army's reason for being is to fight conflicts that are the result of failed politics and diplomacy.
However, assume you are a front line Infantryman, engaged in a combat action. Or a Guardsman doing a MUTA 6 or AT. Doesn't much matter, with regard to the other MOS.
Without Operations support and Logistic Support working together with Intelligence and Administration, with clerks, truck drivers, pilots, crew chiefs, supply folks, medical folks, armament folks, procurement folks, budget folks, finance folks and maintenance folks at all levels, you'd be on the front lines:
Naked
Weaponless
Without Ammo
Without equipment of all types
Without LBE etc.
Without air support
Starving
No fuel
No mechanics
No paycheck
No bandages and medevac
No way home
Every CMF has a series of MOS designed to directly or indirectly support the Combat soldier, thus CS/CSS classification. You many not need a medic or helicopter dustoff or A-10 support today, but tomorrow you may be screaming for them. Or dead without them.
As you progress through the ranks you will become more and more aware of these, and my advice is, start asking questions of your leadership to learn about them. As a leader, you are all of the above, administration, intelligence, operations and logistics. At Battalion/Squadron and levels above, they have S, G, and J sections with Officers and NCOs to perform all aspects of those, and to get what you need to Company/Troop level and below.
In your Company/Troop, the CO is responsible for both of those, and has the XO and 1SG to deal with all aspects of handling everything the soldiers need. At Platoon/Section level, PL are the last level of Officer, and NCO do all of the rest of those functions in one form or another.
If you are in charge of soldiers, YOU are the admin, intelligence/security, operations and logistics guy. You write evaluations, you give input to training schedules, you schedule or conduct inspections, you let the LT/PSG know how many folks you have to insure the proper amount of ammunition, chow, NVD batteries, etc. If your unit fails in it's task(s) YOU are the first line supervisor. You don't have to become an expert on everything the Army does, but you SHOULD know at least the basic functions of the CS/CSS so you will know where to go in the absence of higher leadership.
If you have great tires, awesome paint, and a multi-speed transmission, but no engine, your car goes nowhere. If you have an engine, you can go, but not if the other components are missing. And if your paint job sucks, no one is going to notice the rest.
ALL the moving parts matter, and ultimately, without any one of them, the entire system fails.
So, in my (not so humble) opinion, NO MOS is any more important than any other. ALL have a function to perform, and all depend on the others to perform those functions adequately.

SSG (Join to see) List the 10 most unimportant MOS in order. Then imagine you trying to complete the mission without those MOS qualified soldiers.

I'm surprised no one mentioned Signal. Tsk Tsk!

I don't know why you'd want to make fellow Soldiers from the MOS's that don't make the list feel somehow less.

Infantry is first, all others are support. SMH at this silly question.

Engineers lead the way!!!!!

The infantry may be the queen of battle but the engineers make sure that bitch doesn't get raped

Awesome question for entertainment purposes only!
I was trained that the combat arms series require at a minimum: beans, bullets and bandaids. Not necessarily in that order. We can logically place intelligence, communications, logistics and some form of transportations.
Pay/contracting "could" be civilianized, so I'll skip that since we're just discussing. The military band is fired and replaced by individuals who also play musical instruments or a tape recorder as needed.
Ima have to agree with MSG Dan Castaneda that the 18 series covers at least the shooting, communicating, intelligence and medical. They can train and run up to a BN sized element so I'll list that as #1.
They require support, so riggers and plane/helicopter drivers, logistics, MISO and higher level medical.

COOKS!! Without cooks, nobody eats. And therefore everybody dies. Mission failure.