Why torsion vanishes in supergravity? Why is it (more often than not) considered in supergravity that the torsion vanishes in Cartan's first structure equation? What does the vanishing of the torsion imply? 
 A: In SUGRA, torsion does NOT vanish. You are always left with fermionic torsion terms. The spin connection includes torsion terms in SUGRA that does not vanish
$\omega_\mu{}^{a b} (e, \psi_\mu) = \omega_\mu{}^{ab} (e) + \frac12 \bar\psi_\mu \gamma^{[a} \psi^{b]} + \frac14 \bar\psi^a \gamma_\mu \psi^b$
where $\omega_\mu{}^{ab}(e)$ is the torsion-free spin connection and $\psi_\mu$ is the gravitino. So, quite opposite to what you asked, in SUGRA, the torsion terms do not vanish.
A: Generally, the vanishing of torsion means that spacetime looks in each first order infinitesimal patch like the model spacetime (Minkowski spacetime). One may take this as a mathematical formulation of the principle of equivalence. A nice account of this is in section 3 of 
John Lott, "The Geometry of Supergravity Torsion Constraints" Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125
And least in 11d supergravity it is the bosonic part of the super-torsion that does vanish. In fact its vanishing here is equivalent to the equations of motion! Requiring the full super-torsion to vanish, in 11d, is equivalent to the EOM for purely bosonic solutions.
This is due to a remarkable result by Candiello-Lechner-Howe. For review and further pointers see the PhysicsForums-Insights article:


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*11d Gravity from just the torsion constraint
A: To add to what some of the others have said:  In 10d supergravities at least, the 3-form field strength $H = dB$ can be re-cast as torsion.  But it's usually easier just to say it's a 3-form field strength.
Nonzero fermions do require torsion, but usually when solving SUGRA, one is looking for classical solutions, on which the fermions vanish, and thus there is no torsion.
A: The two main motivations to not consider torsion are:


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*Classical General Relativity (that has zero torsion) nicely fits the experimental data 

*Computationally, it's (more) difficult to deal with a non torsion-free connection.
The two motivations are of course strongly related!
Moreover, a quote from Carroll's book could be relevant:

We could drop the demand that the connection be torsion-free, in which case the torsion tensor could lead to additional propagating degrees of freedom. Without going into details, the basic reason why such theories do not receive much attention is simply because the torsion is itself a tensor; there is nothing to distinguish it from other, "non-gravitational" tensor fields. Thus, we do not really lose any generality by considering theories of torsion-free connections (which lead to GR) plus any number of tensor fields, which we can name what we like.

EDIT (after the comments): In second order formalism of General Relativity, you deal only with a torsion free connection and torsion free spin connection. You can indeed show that the difference with the 1° order formalism (with torsion) is a quartic fermionic term. Just to mention, you can even go through the 1.5 formalism, that is a mix of these two.
It could be helpful cap. 8 of Supergravity, Van Proeyen-Freedman
