What are the pros and cons of Einstein-Cartan Theory? As an alternative to General Relativity, i hear that it can avoid the initial big bang singularity as well as the singularities in black holes, so why does it appear to be talked about so little? If anyone can enlighten me a bit on the successes and shortcomings of it i would be very grateful. 
 A: I'm not a fan of tl;dr-style answers, so here's my extended take:
Some Historical Remarks:
When Einstein worked on his theory of gravity, the notion of a connection was, at least to most, still synonymous to that of a symmetric connection, i.e. in particular symmetric connection coefficients $\Gamma^\alpha_{\mu \nu} = \Gamma^\alpha_{\nu \mu}$. He was later approached by Élie Cartan who at the time investigated more general connections and pioneered the notion of torsion in his famous formalism based on tetrads and connection forms. He published his work as

"Sur une généralisation de la notion de courbure de Riemann et les espaces
à torsion" (1922)

and

"Géométrie des espaces de Riemann" (1925)

and subsequently discussed his findings with Einstein in the context of an extended theory of gravity. Since, as previous answers to this question note, the effects are indeed "very small" (more on that later) and thus limited to events like the big bang and black holes where the spin density is high enough, they discarded their attempts (among other reasons like a missing Schwarzschild solution).
Note however that at the time the concept of intrinsic spin as used in contemporary QFT was not yet prominent.
While these initial attempts to extend General Relativity to a more "natural" or rather holistic geometry, i.e. Riemann-Cartan, arguably failed due to the boundaries of their time, the subsequent success of General Relativity drastically changed the overall climate:
Often in history when a physical theory becomes largely successful, any alternate theories that did not immediately couple to experimental evidence, no matter the possibility to produce adequate ones, become more or less decried as "nonsensical" or "exotic". A famous example would be the "Copenhagen Interpretation" of Quantum Mechanics, which socially and scientifically smothered any alternate approaches for decades, many of which are now experiencing a long overdue renaissance. Based on the reading I have done so far, I highly suspect that linear and quadratic Einstein-Cartan theory suffered a comparable fate albeit on a smaller social scale.
While the baffling success and novelty of GR stands second to few, it has some obvious flaws that arise on both cosmological and microscopic scales. I will touch on them during the pro/con list. For a rather recent inspection of the current standing of EC theory I recommend this discussion on researchgate and the papers I will mention below.
Mathematical Complexity:
I'll state very boldly that from experience the formalism does not change significantly and neither does the mathematical complexity. Yes there is an additional tensor field, the torsion, and depending on the formulation you might see the contorsion pop up, but the beauty of Einstein-Cartan theory lies in the lack of exotic additions to the action both mathematically and in overall form. The action remains an integral over the curvature which now can be split into the usual curvature of the Levi-Civita connection and contributions due to non-zero torsion.
Formulated in the language of tetrads, a beautiful review of which can be found in this introductory article, the action is defined as
$$ S_{EC} := \int_M tr\left[ F_\omega \wedge e \wedge e \right],$$
which superficially is precisely the same definition as that of the Einstein-Palatini action and hence common GR. The only difference is that the torsion defined as
$$ d_\omega e = de + \omega \wedge e, $$
is not required to vanish. If one includes a source Lagrangian $L_M$, the equations of motion are:
\begin{align}
    F_\omega \wedge e &= \frac{\delta tr[L_M]}{\delta e}, \\
    d_\omega e \wedge e &= \frac{\delta tr[L_M]}{\delta \omega}.
\end{align}
Note that the first equation represents the usual Einstein field equations while the second one is the additional equation for torsion. I'll comment on the interpretation of both further down. For now note that mathematically the language and the complexity does not change, even when transferred to the more common index-notation. There are additional equations and there is an additional tensor field, however the same can be said about almost any attempt to extend GR and thus should not be seen as a problem of EC theory.
Meanwhile from the mathematical point of view it is only natural to assume a connection, and GR is nothing but a physical theory of the Levi-Civita-Connection, has both curvature and torsion. While there also exist other interesting generalisations, EC theory is without a doubt the most natural extension of GR from a mathematical perspective.
Cons and Pros:
In the following I will cite mostly from this article from 2013 by Nikodem J. Poplawski, who has written a series of articles on EC theory and its effects on cosmology and QFT. This article is a particularly good read since it combines a lot of work on the field and is a quick read with many good references to go on.
Cons:

*

*Spin-torsion coupling only becomes relevant at extremely high mass (and thus spin) densities. Poplawski claims (page 2) that for a neutron the "Cartan density", i.e. the density for which torsion effects become relevant, is of order $10^{45} kg/m^3$. This corresponds to densities reached only at the big bang and potentially inside black holes. Thus a direct observation of spin-torsion coupling is well out of reach for current experimental physics.

*Torsion has an algebraic and linear differential equation and thus vanishes completely in vacuum. While this is also true for the Ricci curvature, even in common GR, this does not imply that the Riemann curvature tensor vanishes, hence effects of GR can propagate through vacuum. An analogue tensor does not exist for torsion in EC theory and many argue that this is highly restrictive for a physical theory. A popular attempt to solve this is adding a quadratic torsion term to the action which results in one mixed equation for curvature and torsion. However a sharp decay of such propagation outside matter has been derived for interactions with massive scalars and spin-1 bosons in this paper.

*Einstein-Cartan theory is not the minimal theory that explains the effects of GR. This is partly due to the fact that EC is GR whenever in spin-vacuum. It is thus mandatory to look for additional effects and how we can observe/probe them, but some of the proposed ones lie far from our current scope.

*Torsion adds a cubic spinor term to the Dirac equation, compare equation (5) of this paper. Indeed the self-interaction term, containing only the metric and spinors, appearing in the action yielding this modified Dirac equation is not renormalisable, however the Lagrangian as a whole is since it includes torsion.

*EC theory does not comply with the popular view of the universe being of finite age. While this leans into the Pros, it postulates a big-bounce solution to the cosmic expansion and a lot of room for arguably speculative interpretations thereof.

Pros:

*

*EC seems to give simple answers to several problems of cosmology without exotic matter, an inflation field or singularities. Many of these can be found in Poplawski's article from 2013. While some of his ideas are speculative, he shows a clear silver lining to many long-standing questions of cosmology and provides ample evidence that torsion could be the answer.

*EC can be naturally formulated as a gauge theory of the Poincaré group. This is done in the introductory article mentioned above and in this article which also covers many related topics. Note that this is the language of contemporary gauge theory, thus one can employ the apparatus of quantisation more easily.

*Torsion is a part of gravity already. There exists an equivalent formulation of GR using a connection without curvature but non-vanishing torsion. It is called Teleparallel Gravity (TEGR) and was pioneered by Einstein himself when he tried to unite GR and Maxwell's theory of electromagnetism. In this theory torsion takes the role of a force in a flat geometry providing the same evolution as GR. I highly recommend you check TEGR out, if you want a holistic geometric picture of EC.

*Intrinsic spin has no analogue in classical GR. This is addressed in EC via the natural spin-torsion coupling that provides many interesting effects at high densities like the avoidance of singularities and the Hehl-Datta equation. It also provides a natural derivation of the conservation of total spin. All in all this makes EC a promising candidate for advances towards a theory of quantum gravity.

*There exist many approaches to solve even further problems of contemporary physics using EC and extensions thereof. Among them are Dark Matter, the Black Hole Information Paradoxon, the rotation curves of spiral galaxies and a possible preferred direction in the universe. Note that this is an active field and while there are many interesting papers, some of them remain speculations due to the problems mentioned above (conveniently I can't put more links in this answer).

Conclusion:
EC is an interesting and natural extension of Einstein gravity that certainly deserves more attention than it has gotten over the majority of the past century. While many of the answers it proposes for long-standing questions of physics sound promising, they remain partly elusive in the face of current experimental possibilities.
It will likely be the touchstones of Quantum Gravity and Cosmology that ultimately decide if Einstein-Cartan theory or a portion thereof is realised in nature. Thus it remains the upmost goal of any advocate of EC, theoretical or experimental, to provide meaningful predictions beyond GR and explore the role of torsion in unifying GR and Quantum Mechanics.
If I failed to write according to the etiquette of this forum, let me know and I will edit my answer accordingly. The same goes for possible additions, mistakes and the like.
A: A con i know: The Dirac equation becomes nonlinear and therefore the superposition principle used for the quantisation doesn't work anymore. 
But it should be mentioned, that the difference in predictions is so little different from GR that nowadays we aren't able to measure which one is "correct".
A: One main reason why the ECKS theory (Einstein-Cartan-Kibble-Sciama) is so unpopular even today is its extreme mathematical complexity.  The connection isn't symetric anymore, so it changes a lot of things in the mathematical formalism (for example: the covariant derivative applied to a simple scalar field doesn't commute anymore):
\begin{equation}\tag{1}
\Gamma_{\mu \nu}^{\lambda} \ne \Gamma_{\nu \mu}^{\lambda}.
\end{equation}
The non-symetric connection implies the existence of a tensor field called torsion, which is a three indices tensor field:
\begin{equation}\tag{2}
T_{\mu \nu}^{\lambda} \equiv \Gamma_{\mu \nu}^{\lambda} -\Gamma_{\nu \mu}^{\lambda}.
\end{equation}
This tensor field could be extracted everywhere and added to the Lagrangian as an extra field, so most people prefer to use standard GR (with a symmetric connection and standard formalism), and simply add the new tensor field by hand in the Lagrangian.
You may explore the Wikipedia page, but beware: it has a few subtle mistakes:
https://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory
So PROS:  It's a natural extension of General Relativity.  GR feels more "complete" with spacetime torsion.
CONS:  It's terribly more complicated than GR without torsion. Torsion occurs only inside matter, in cases of extremely high density states, so it's very hard to test it in labs!  Propagating torsion outside matter is probably too weak to be observable at all (if it could propagate outside at all!).
