Would singlet oxygen in the engine cause a more efficient combustion?
This hypermileage scheme is one of the more interesting ones I've come across, so I think it's worth some discussion. I don't know much at all about the engineering aspects of internal combustion engines, so I'll stick to a theoretical analysis.

Let's start by describing the difference between singlet oxygen and triplet oxygen. Both of these are the same substance, molecular (di)oxygen, or $\ce{O2}$. However, they are in different energetic states. In its lowest energy state, a molecule of $\ce{O2}$ contains two unpaired electrons (which is a very interesting and important fact in itself), and this form is called triplet oxygen. This is the regular stuff you breathe all the time.

However, if some energy is given to triplet oxygen, it can be excited into a different state, and the two formerly unpaired electrons now pair up. Since there are now no unpaired electrons, we call this singlet oxygen (technically there is more than one singlet state possible, but for our purposes that can be ignored). 

So in summary, singlet oxygen and triplet oxygen are the same molecule, except that singlet oxygen has inherently more energy. So why does this matter? If singlet oxygen has more energy, then one expects it to be more reactive. It is entirely conceivable that a fire supplied with singlet oxygen will burn hotter/more completely than the same fire supplied with triplet oxygen. So that part of the scheme is plausible.

That said, you may ask yourself, if all it takes is some energy to turn triplet oxygen into singlet oxygen, then shouldn't oxygen in any fire be continuously converted from the triplet state to the singlet state, simply by the heat of the roaring flame? This would mean preparing singlet oxygen separately is entirely superfluous! As it turns out, due to the quirks of quantum mechanics it is not easy to excite triplet oxygen into singlet oxygen. Technically speaking, this is called a spin-forbidden process.

However, due to special relativity (!), spin-forbidden processes turn out to not be completely forbidden, just very unlikely. A concept called spin-orbit coupling comes into play. The more spin-orbit coupling in a chemical system, the more likely it is for spin-forbidden process to nevertheless happen. Spin-orbit coupling becomes more and more relevant as you go further down the periodic table. This means heavy atoms such as platinum can favour the transformation of triplet oxygen into singlet oxygen. So again, the necessity of a metal honeycomb as pictured in the scheme is at least conceptually plausible for this singlet oxygen strategy to work.

(Of course, it is possible to produce singlet oxygen through other means, such as the decomposition of some oxygen-containing molecules like hydrogen peroxide and ozone, but doing so would defeat the whole purpose; you'd essentially have to buy a second kind of fuel!)

At this point it seems there is actually something going for this scheme, but here is where so many others like it stumble: the laws of thermodynamics. If it takes energy to transform triplet oxygen into singlet oxygen, then that energy has to come from somewhere. If the energy comes from the metal honeycomb itself (or more precisely, the energy imbued into the metal honeycomb during its manufacturing process), then yes, it is plausible to get increased mileage, but the mesh will lose energy until it is inactive and has to be replaced or reactivated. My bet is that this would cost much more than the meagre fuel savings.

In principle, the energy to make singlet oxygen can come from the extra energy from the more efficient fuel burn, leaving the mesh intact. However, now you're dipping into the very extra energy that was supposed to make the car get more mileage. The detailed energy balance is generally very hard to calculate, and I couldn't begin to tackle all of the real-world variables in play. That said, the answer to whether a hypermileage scheme is cost-effective almost always ranges from a flat-out "no" to "yes in principle, but no in practice". Processes that are cost-effective (precise spark timings, regulation of air-fuel ratio, etc) tend to make it into standard issue vehicles in the first place. I can't help but be reminded of the joke that alternative medicine which works is just called medicine.

Lastly, regardless of energy savings, singlet oxygen may not be great inside the engine. Not only is it more chemically reactive than triplet oxygen, but it can perform reactions which are completely different from triplet oxygen. It could cause more wear of the engine components, or could generate more of some unwanted sideproducts during combustion, such as nitrogen and sulfur oxides. Any singlet oxygen that manages to escape combustion and exit the tailpipe could also pose a significant biological and chemical hazard.