who out ranks who? a male drow noble, or a drow female commoner?

This should be in Story & Lore, as there is no rule on this. 

From the Monster Manual:

"In drow society, males are subservient to females. A male drow might lead an Underdark patrol or a raiding party to the surface, but he reports to a female drow — either the matron of his house or one of her hand-picked female subordinates. Although male drow can fill almost any function in drow society, they can’t be priests, nor can they rule a house."

This is only relevant to the default Forgotten Realms. If your campaign is set anywhere else check with the DM. Even if it is FR still check anyway because the DM is entirely free to change this.

Quote from Cyb3rM1nd

From the Monster Manual:. Although male drow can fill almost any function in drow society, they can’t be priests, nor can they rule a house."

 

 brigan de'arthe had a leutentant who was a male drow, multiclassed wizard and preist.

Quote from Banren_Cunundrum >>

Quote from Cyb3rM1nd

From the Monster Manual:. Although male drow can fill almost any function in drow society, they can’t be priests, nor can they rule a house."

 

 brigan de'arthe had a leutentant who was a male drow, multiclassed wizard and preist.

 
Which is a character made by authors who are free to change whatever they want.

For your own games, check the DM.

Any DM, author, etc can make whatever exceptions they want. There is no official "it must be this way". 

In typical Drow society, a male drow from a noble house is higher than a female drow comoner, except if said female is a priestess of Lolth in a Lolth proiminent city.

It doesn't matter: neither is in a position to give commands to the other. It's like asking "who outranks who, an intern at a law firm or a manager at McDonald's?"

Commoner woman outranks noble man, if it's just a question of who holds the door for whom.  However, if a commoner woman presumed to give orders to a man of a noble house, she would then get to explain why to the noble women of his house.  Unusual individuals, like celebrities, skilled artisans, beloved companions, can get away with more.

In general, drow men are explicitly subservient to drow women. I wouldn't expect a male from a noble house to be expected to follow any direct commands from a female commoner though, if only because the women he serves directly (and more or less belongs to) outrank her. Any interaction would have a dynamic of the commoner and the servant or slave of a noble, and drow society tends to be very hierarchical so she would be unwise to do anything that his mistress(es) might consider uppity, pretentious, or otherwise offensive to themselves by proxy. 

Flush master and others have the basic idea, Drow males can be clerics, but not of Lolth. They are typically hidden clerics of Vhaeraunn so they typically multiclass to cover it.

If you look at real societies with strict gender roles mixed with other hierarchies, you will see that regardless of whether class trumps gender or gender trumps class, class will trump gender in actuality. Even if the status of a noble male was officially lower than a commoner female, the noble male would usually have influence with noble females, either a mother, a sister, a daughter, or a lover, who would intercede for the noble male against a commoner female, at least privately.