My understanding is that the particle is a somewhat artificial notion in QFT (see: Quantum Mechanics: Myths and Facts), and that in general it is possible for a quantum field to have unstable excitations that don't look anything like particles. Is this an active field of research (what is it called)? Are there experimental searches for detection of such non-particles? For example, could dark matter be non-matter: some large-scale unstable oscillation of a quantum field?
There is nothing artificial about particles as quanta of a field. Each finite-energy configuration of a quantum field may be written as a complex linear superposition of $N$-particle states with various values of $N$. A general and generic state in the Hilbert space isn't an eigenstate of the "number of particles operator" but that's true for any other observable, too: most states in the Hilbert space aren't eigenstates of a predetermined operator.
This is not a "problem"; it just means that if the observable expressed by the operator is measured, one may get different values as the result of the measurement. The probabilities of individual results are calculated using the standard quantum formula – the Born rule – as the squared absolute values of the complex probability amplitudes (inner products of the state vector with an eigenstate etc.).
In nonlinear field theories, one may often write down classical solutions called "solitons" which are stationary yet localized; there also exist quantum states in the Hilbert space whose support is close to the classical solution. Magnetic monopoles are a good example. Strictly speaking, the quantum states corresponding to these solitons may still be formally written down as combinations of the usual $N$-particle states but this way of writing them becomes useless because the nonlinearities in the equations of motion for the fields totally change the expected behavior relatively to a free field theory for which the $N$-particle-state basis is most useful.
If the fields are only excited by field modes with a long (macroscopic) wavelength, the interpretation in terms of ordinary particles becomes – in contrast with your expectations – most appropriate. The long wavelength is interpreted as the particles' having a very low momentum.
There is no "active field of research" of the type you suggest. Instead, the field you are describing is "learning the first classes in a basic undergraduate course of quantum mechanics". For example, if the dark matter is composed of neutralinos, they're the excitations of an ordinary fermionic field $\chi(x,y,z,t)$ transforming as a spinor, and the basis with $N$-particle states of several neutralinos with different momenta is a perfectly valid basis of the whole Hilbert space and it is therefore enough to describe anything that may physically occur to this field.
I have mentioned solitons. Dark matter could perhaps be made of solitons except that I am not aware of any viable models of this kind.
This is a high energy theory which extends the standard model by an additional scale invariant sector of particles whose properties such as energy, momentum, and mass can simultaneously be scaled up or down (therefore the term scale invariant). In the standard model, these would only work for photons which are massless.
These new particles, if there are expected to coupling only weakly with "normal" matter at low observable energy scales and behave some kind of similar to neutrinos. At the LHC, such unparticles would for example become noticeable by missing energy. There are indeed ideas, that dark matter could be made of such unparticles, see for example this paper.
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