Higgs Boson Decays Still Well Behaved UPDATED

In short, six years after the experimental discovery of the Higgs boson, the particle detected so far still shows exactly the decay properties that it is predicted to have in the Standard Model for a mass of the magnitude observed.

A recent CMS experiment report from the Large Hadron Collider (LHC) finds that overall Higgs boson branch fractions (i.e. the relative proportions of decay products of Higgs boson decays) are as follows:

The best-fit ratio of the signal yield to the standard model expectation is measured to be μ=1.17±0.10, assuming a Higgs boson mass of 125.09 GeV. 

A recent paper from the ATLAS experiment at the LHC has yet to find anomalies in Higgs boson decays and instead can merely quantify the bounds of the maximum possible anomaly sizes that can be excluded at the two sigma level:

As the only seemingly fundamental scalar in the Standard Model (“SM”), associated with an all-permeating, ever-present field which generates the mass of all SM particles, the discovery of the Higgs boson has opened a new window through which nature can be probed. While initial studies into its properties are compatible with the SM predictions [1, 2, 3, 4, 5, 6, 7], further studies are required to establish a full understanding of the nature of the Higgs sector. Two such avenues are searches for rare decays and beyond the SM (“BSM”) decays of the Higgs boson, with the ATLAS detector [8] at the LHC [9]. These searches share two common themes. First, these are at best rare, or potentially non-existent processes. No significant excesses have yet been observed. As such 95% confidence level upper limits (“limits”) are set according to the CLs prescription [10, 11], using maximum likelihood fits (“fits”), sometimes to multiple variables simultaneously. Second, due to the low expected yield of these processes, the dominant uncertainty for most of the analyses shown is due to limited data statistics. . . . 

While no searches to date have resulted in an observation of a significant excess, they are all statistically limited, and so should improve with the square root of the delivered integrated luminosity. This is especially promising looking to the High-Luminosity LHC, where for example, the limit for the H → J/ψγ search is set to improve to ∼ 15 × σSM [31]. 

The search bounds so far on rare decays predicted by the Standard Model are roughly a factor of roughly 100 in most cases. In other words, we have experimental confirmation that decays that are predicted to have a frequency of not more than X have an experimentally determined frequency of less than 100X, which honestly, is a pretty ho-hum result. 

The LHC is sensitive to a specific kind of initial BSM decay products in the mass range of 15 GeV to 60 GeV.  Backgrounds overwhelm signals for higher masses and experimental apparatus limitations limit detection of lighter decay products. These limitations are so far, however, not very tight ones:

As collider energies increase, analyses often search for new higher mass states. However, new states could also exist at lower masses, undiscovered by previous experiments, if their only significant coupling is to the Higgs boson, which has only been produced in a sufficient abundance at the LHC. Therefore, the discovery of the Higgs boson unlocks a new range of searches for low mass particles predicted by various BSM models. For many of these models, the tightest constraint comes from fits to the major bosonic decay channels of the Higgs boson (H → ZZ & γγ), which provide model-dependent indirect constraints on the branching ratios of ∼20% [21].

Null results individually aren't very exciting, but collectively, they provide robust support for the Standard Model of Particle Physics as an accurate description of the laws of Nature.

UPDATE October 8, 2018:

New results greatly narrow the room for non-Standard Model Higgs boson couplings.

We perform global fits of the Higgs boson couplings to all the 7 TeV, 8 TeV, and 13 TeV data available up to the Summer 2018. New measurements at 13 TeV extend to include the Higgs signal strengths exclusively measured in associated Higgs production with top-quark pair and the third-generation Yukawa couplings now have been established. Some important consequences emerge from the global fits. (i) The overall average signal strength of the Higgs boson stands at 2σ above the SM value (μ=1.10±0.05). (ii) For the first time the bottom-quark Yukawa coupling shows a preference of the positive sign to the negative one. (iii) The negative top-quark Yukawa coupling is completely ruled out unless there exist additional particles running in the H-γ-γ loop with contributions equal to two times the SM top-quark contribution within about 10%. (iv) The branching ratio for nonstandard decays of the Higgs boson is now below 8.4% at the 95% confidence level.

Kingman Cheung, Jae Sik Lee, Po-Yan Tseng, "New Emerging Results in Higgs Precision Analysis Updates 2018 after Establishment of Third-Generation Yukawa Couplings" (October 5, 2018)