The practice of using Gaussian copulas in modeling credit derivatives has come under a lot of criticism in the past few years. What are the major arguments against using the copula method in this respect?
The limitations of the Gaussian copula were well-known among the quantitative finance practitioners before the crisis. See this paper by D. Brigo.
To answer the question:
- no "fat tails"
- unable to fit the market prices without tweaks (base correlation) which make the model arbitrageable
- it's a static model (e.g. forward-starting tranches are impossible to price -- but nobody trades them now anyway)
This said, all other models are either worse or offer cosmetic improvements. Changing the Gaussian factors to some others doesn't really give you much. A few years ago the Random Factor Loading model was en vogue, but it turned out to be much harder to calibrate, and still not flexible enough.
If you want a 'pop science' account for it, the Wired article by Felix Salmon is a pretty good start.
If you want harder technical stuff, well then you can start at the Wikipedia article and its section on Applications and follow the references:
[...] Some believe the methodology of applying the Gaussian copula to credit derivatives to be one of the reasons behind the global financial crisis of 2008–2009. Despite this perception, there are documented attempts of the financial industry, occurring before the crisis, to address the limitations of the Gaussian copula and of Copula functions more generally, specifically the lack of dependence dynamics and the poor representation of extreme events. The volume "Credit Correlation: Life After Copulas", published in 2007 by World Scientific, summarizes a 2006 conference held by Merrill Lynch in London where several practitioners attempted to propose models rectifying some of the copula limitations. See also the article by Donnelly and Embrechts  and the book by Brigo, Pallavicini and Torresetti .