I am currently trying to get to grips with MHD instabilities in fusion experiments, and I am confused at to why instabilities are sometimes referred to as modes? To my understanding, a 'mode' is a particular solution of the wave in the torus given a set of conditions, and instabilities can occur at a wide range of modes - so why the use of 'Kink mode', for example?
As a follow up, why do so many limiting conditions (eg the beta-limit) focus on the (1,1) mode, with no discussion of the rest?
Yes, 'mode' is a particular solution of the MHD equations for a given set of conditions. Usually, these solutions assume the perturbations are negligible and that the nonlinearities are small. The mode solution typically leads to $\partial F_0/\partial t = 0$ where $F_0$ is an independent variable. Now this solution may or may not be stable which means it must be tested for stability by checking if small perturbations die out (stable) or they grow (unstable). To do this we assume $F= F_0 + \delta F$. While $\partial F_0/\partial t$ is still zero, $\partial \delta F/\partial t$ may not be.
When the perturbations (i.e magnitude of $\delta F$) are likely to grow, it is called an instability otherwise it's just a stable mode.
As a follow up, why do so many limiting conditions (eg the beta-limit) focus on the (1,1) mode, with no discussion of the rest?
Because it was, by far, the most annoying problem in early machines. It overpowered all of the other instabilities to the point where they were invisible. When ZETA was seen to suppress M=1 they really thought they had cracked the nut. That's when they started noticing the higher-order wave-driven problems.
It should be noted that different designs, like the mirror, had entirely different instabilities and these sorts of rules-of-thumb are different. A lot of the reason you see (1,1) is simply that those designs have largely become historical and they don't talk about the issues they had in modern works.
An instability is the mechanism or pathway through which energy is exchanged between the free energy source (e.g., unstable particle velocity distribution function) and the electromagnetic fields.
The modes are usually the ever-present thermal oscillations of the system that can accept the transfer of energy through the instability to grow and electromagnetic fluctuation of some kind.
You can think of the modes as channels for a walkie-talkie and the instability is the walkie-talkie.
Unfortunately, these terms are often used synonymously in the literature but they aren't really the same phenomena.
I explain a little more in https://physics.stackexchange.com/a/517523/59023.
