Toy models of asymptotic safety? Are there some toy model QFTs where the asymptotic safety scenario is realized? 
 A: If by asymptotic safety you mean theories that have an interacting UV fixed point, they're a dime a dozen in field theory. Just pick any conformal field theory you like and see if it contains a relevant scalar operator. If so, deform away from the fixed point with that operator. From the viewpoint of wherever it ends up in the IR, the UV is "asymptotically safe": the RG flow approaches an interacting fixed point. Not every CFT has a relevant scalar operator, but they're not uncommon, especially if you don't demand that they be singlets under all the global symmetries. For instance, start with N=4 supersymmetric Yang-Mills theory and turn on masses for some of the scalars and/or fermions. You end up with some theory in the IR which is "asymptotically safe."
This answer is a bit of a cheat, because the context where people usually talk about asymptotic safety is gravity, and whatever they mean by "fixed point," it surely isn't a conformal field theory. We know that gravity is not a local field theory: it has no local observables and high-energy scattering creates large black holes, which means the states of the theory look very unlike the states of any local quantum field theory. Furthermore, Weinberg and Witten proved that if you start with an ordinary four-dimensional field theory with a conserved stress-energy tensor, you will not find a massless graviton in the spectrum of your theory. So if gravity could be asymptotically safe (personally, I'm pretty confident that it can't), the "UV fixed point" is not an ordinary quantum field theory fixed point. I've never heard anyone propose anything very concrete for what it is, so in the context of gravity "asymptotic safety" always seems like a very fuzzy concept to me.
A: There are a number of preprints discussing this. I'll just list out the ones I've read:


*

*Sandor Nagy,
"Lectures on renormalization and asymptotic safety".
Annals of Physics 350 (2014), pp. 310-346.
Eprint arXiv:1211.4151
DOI:10.1016/j.aop.2014.07.027

*Jens Braun, Holger Gies, Daniel D. Scherer,
"Asymptotic safety: a simple example".
Phys.Rev.D 83 (2011) 085012.
Eprint arXiv:1011.1456
DOI:10.1103/PhysRevD.83.085012

*J. Kovacs, S. Nagy, K. Sailer,
"Asymptotic safety in the sine-Gordon model".
Eprint arXiv:1408.2680, 8 pages.

