Who found the Ising transition? The famous story is that Ernst Ising studied the 1d classical stat mech model which bears his name, argued it has no phase transition, and guessed that the same would hold in all dimensions. He was wrong of course, but who actually found the phase transition in this model?
 A: The first proof of the existence of a phase transition in (the two-dimensional version of) this model is due to Peierls in this classical paper from 1936. The method of proof, today known as Peierls' argument has become a cornerstone in the rigorous analysis of phase transitions and been the source of many generalizations of extremely wide applicability.
A few years later, exploiting the self-duality of the nearest-neighbor Ising model on $\mathbb{Z}^2$, Kramers and Wannier determined (heuristically) the value of the critical temperature in this paper from 1941. Many ideas introduced in this paper have also led to important generalizations and further insights later on.
Finally, the first computation of the free energy of the planar model (at zero field) was made by Onsager in his celebrated work in 1944 and allowed an analysis of the critical properties of the model. The implications of this work triggered the whole modern theory of phase transitions.

Concerning the rigor of Peierls' work: in my opinion, all the ideas in the paper are correct and the proof is essentially correct (at the mathematician's level of rigor; of course, it's much more rigorous than many works in theoretical physics). Nevertheless, the first fully rigorous versions of his argument are usually considered to be these papers by Griffiths and Dobrushin.

Finally, concerning the history of this model and of its impact on modern statistical mechanics, I strongly recommend this series of papers by Niss : 1920-1950, 1950-1965, 1965-1971.
