Mountains' role as "Earth stabilizers" isn't scientifically confirmed.. yet

 Q 16:15 "He has made mountains stand firm on the earth, to prevent it shaking under you"  

Modern Muslims shouldn't claim a scientific miracle here, since the science isn't there yet.. and because of the difficulty of 'removing a mountain' to check the effects, it may never be confirmed.  
The usual rationalization of the ijaz ilmi people goes like this: it prevents earthquakes (not proven. Actually mountainous regions are earthquake-prone), or it keeps the planet's rotation stable by making the surface not too smooth (by friction against what exactly?!), or by stabilizing the crust by pinning it to the mantle.  
This last one became popular, and they use terms like "isostatic balance" without fully understanding them.  

Isostasy is about the mountain stabilizing *itself*, reaching a state of balance, like a floating iceberg. The science says nothing about it functioning as stabilizers for its surrounding!  
This point can't be emphasized enough. It's acting on itself, not on the land around it, at least as the state of geology & physics goes today.  
I do believe the Qur'anic claim, that mountains are pegs that stabilize the surface of the earth, but that's faith-based belief, not seeking scientific approval.  

The Arabic word used isn't "shaking" btw, it's m-i-d, which is closer to "sway". It's a nautical term too, used to describe the role of ballasts on a ship, i.e. weights that stabilize its swaying motion. If you take that as referring to the planet's motion in space (I don't) then it's about balancing weights arranged on the surface to keep the motion regular. This is wrong, because compared to the mass of the planet itself, mountains are nothing! The whole crust (mountains included) is as thin as an apple's peel. Earth's radius is more than 6 thousand kilometers! The highest mountain is nothing to this.  
So we take it as referring to stabilizing the area surrounding a mountain. AFAIK this isn't confirmed yet, and it's dishonest to claim it's.

See Q 78:7 , 21:31 , 31:10