For CFT there are many examples. I will give some examples of local conformal nets on the circle (or real line).
The Ising model Pieter mentions is the Virasoro net with $c=1/2$. The Virasoro net can be constructed for the discrete $c<1$ and $c>1$. See eg.
- Kawahigashi Y. Longo R. (2004) "Classification of local conformal
nets. Case c<1" Ann. of Math. 160, p493-522
They furthermore classify all local conformal nets with central charge $c<1$.
Positive energy representations of loop groups give conformal nets.
- Jürg Fröhlich and Fabrizio Gabbiani. Operator algebras and conformal field theory. Comm. Math. Phys. Volume 155, Number 3 (1993), 569-640. Link
The conformal nets associated to lattices and its orbifolds are constructed in
- Dong & Xu. Conformal nets associated with lattices and their orbifolds. Advances in Mathematics (2006)
Volume: 206, Issue: 1, Pages: 279-306
and in the same issue Kawahigashi and Longo have constructed the "moonshine" net.
- Kawahigashi & Longo. Local conformal nets arising from framed vertex operator algebras. Adv. Math. 206 (2006), 729-751.
For massive models in 2D Lechner constructed the factorizing S-matrix models in which are a priori just "wedge-local" nets but he managed to show for a class that to show the existence of local observables.
- Lechner. Construction of Quantum Field Theories with Factorizing S-Matrices. Commun.Math.Phys. 277, 821-860 (2008)