The most standard argument for triviality is the Landau pole of QED and $\phi^4$ theory. This was a valid argument in the early dayys of renormalized QFT but it is no longer valid today. After all, QCD also has a Landau pole, and unlike QED, its Landau pole is at physically realizable energies. See, e.g., p.27 of https://arxiv.org/pdf/hep-ph/9802214.pdf. But nobody ever suggested that this is a reason for the triviality of QCD. Similarly, quantum Yang-Mills theory has a Landau pole, https://arxiv.org/pdf/1311.6116.pdf, but it is believed that the theory exists rigorously, and a proof of it would solve half of the 7th Clay Millennium Problem (the second half being a proof of a mass gap for it).

The Landau pole is most likely an artifact of perturbation theory, caused by the latter's lack of accounting for terms with an analytic dependence of the form $e^{const/\alpha}$, where (for QED) $\alpha$ is the fine structure constant. Evidence for this is the fact that the Landau pole in 1-loop perturbation theory for QED is cancelled exactly by a second, nonperturbative term that does not contribute perturbatively. As a result, the spectral function is well-defined and satisfies the causality conditions required by the Källen-Lehmann representation. See the paper by Bogolyubov, Logunov and Shirkov, *The method of dispersion relations and perturbation theory*, Soviet Physics JETP 37 (1960), 574-581. There is a highly cited review paper of the resulting analytic perturbation theory (concentrating on QCD, though) by Shirkov and Solovtsov, *Ten years of the analytic perturbation theory in QCD*. Theoretical and Mathematical Physics *150 *(2007), 132-152. An interesting recent paper on the subject (for QCD) is https://arxiv.org/abs/1411.2554.

A 2017 paper by Djukanovic et al. shows that treating QED as an effective theory by adding an additional Pauli interaction term already removes the Landau pole at 1-loop order.

For $\Phi^4$ theory, my positive evidence is somewhat limited, since this is somewhat aside from my main goal, to prove the existence of QED.

My main argument for $\Phi^4$ theory is Section 8, ''IS DESTRUCTIVE FIELD THEORY POSSIBLE?'' in a paper by Gallavotti and Rivasseau from 1984, which discusses $\Phi^4$ theory in 4 dimensions and is very pessimistic that the existence of a rigorous $\Phi^4_4$ quantum field theory can be ruled out. The main reason given is that one needs arguments valid for all possible constructions, while the usual approach (leading to negative evidence) is to consider the simplest classes of approaches only. As far as I can tell no work at all has been done since then on their Super-strong Triviality Conjecture, for which they state ''we do not see at the moment any compelling reason to believe it at all''. (This is confirmed by a remark of Gallavoti at the end of p.14 in a 2014 paper:

the conjecture that it is impossible to obtain nontrivial Schwinger functions in a scalar quantum field theory in dimension 4 is still (wide) open

Callaway's 1988 comprehensive review article on triviality comments the first of these papers on p.290 as follows:

Other attempts to construct a nontrivial $\phi^4$ theory are often a bit more abstract in nature. Rigorous discussions of triviality (see section 2) often require that a $\phi^4$ field theory is defined as an infinite-cutoff limit of a ferromagnetic lattice theory. It has been argued [4.31] that this is an assumption whose removal changes the nature of the problem dramatically. Indeed, no argument appears to prevent the existence of an interesting nontrivial ultraviolet limit of an antiferromagnetic lattice $\phi^4$ theory, even in d >4. This remains an interesting open problem.

While this is no direct positive evidence it is doubly negated evidence, which may count as positive, too. Klauder claims that an approach based on affine coherent states should work in any dimension, but his arguments are not rigorous.

See also my answer in the thread ''What is the status of the existence problem for scalar QFT and QED?'', which contains additional information.