Private black ops that does stuff that state-level actors find awkward to do, like retiring former KGB members from their executive roles in a kinetic fashion, trying out at least a dozen different low-to-medium cost approaches until it's done. Some of the money would be needed to decentralize enough to avoid getting squished by regimes who see the writing on the wall.
Fund medical research for things that have no profit motive for pharmaceutical companies, such as rare diseases, psychedelics, herbal remedies or off-patent medications.
This is a longshot, but -- I have a lychee tree that's fruiting right now, with more fruits than I know what to do with, and I'm about to go on a road trip from Florida to the midwest in a week or so. I could drop off a bucket along the way. Shoot me an email at concat("ksymph0", "@", "proton.me") if you're in the area.
Maybe hire a team of architects, engineers and machinists to build machines that automate the building of "earth ships" and subterranean farms, ranches, homes, stores, roads, water storage and treatment, sewage treatment, walking and biking tunnels, light bending and directing mirrors / lenses under ground in places that will become too hot to survive. Each living unit must be modular and expandable yet engineered to survive significant asteroid impacts and last millions of years.
With time some of the tunnels would extend out below the oceans to further increase the probability that humans could survive a blanket of large asteroids and comets with the side effect of negating the usefulness of nuclear weapons thus nuclear weapons could be dismantled and repurposed in mini-micro-mini-micro subterranean nuclear power stations interlinked into neighborhood meshes. Every neighborhood would have a few dozen coolant filled clay heat batteries that would bring every home, road and walkway to 65F (18C) with fresh filtered air from the surface and from air storage geothermal tunnels.
With time the equipment and automation would sufficiently advance to move faster and more efficient. One trillion dollars will not be enough to complete the jobs, but rather to make an optimized blueprint so that every country can repeat and improve upon the process until the entire surface of the planet could be used to produce food until such a time that it has to be produced under ground. Anyone passing by the Earth should be convinced that the entire civilization is agrarian and non-technical thus not a threat. If the tunnels are deep enough and the tunnels are shielded enough from space then fiber networks could share a backup GPS network to reduce dependency on low powered space transmitters.
I believe this would be much easier to accomplish than moving parts of our civilization to the moon and especially more practical and safer than Mars.
I don't think people fully grasp the scale of a trillion dollars.
A trillionaire is a million millionaires.
It's impossible to efficiently manage that much wealth in one brain. Distribute it!
I'd run an essay contest. Every year the best thousand ideas are selected by yours truly and get funded at $100 million each. Even then, it would take a decade to burn through one trillion. It's a tough job, but I accept.
What's interesting about your question is both the variety of answers, but equally how many of them are unimplementable.
Probably the easiest to grasp, and (in theory) the easiest yo do is "give it away". But even this is much harder to do, and more complicated than it seems.
For example injecting substantial cash into a local economy leads to inflated prices. For example revitalized neighborhoods (gentrification) leads to higher housing prices, driving out the local population.
Aid to Africa has shown that in many cases the aid was harmful - inhibiting the growth of local food production etc.
If you choose to make say a thousand billionaires, then expect a billion applications. Good luck deciding.
Even building infrastructure (like say solar farms or nuclear plants) would greatly impact other players in that space. And ending up with all that energy production in one set of hands would be dangerous.
Spending money well is really really hard. Much, much harder than we imagined. Expect every grifter, scammer, organised crime etc to gravitate to your orbit. Filtering value from scam becomes hard.
Bill Gates is working on eradicating some diseases via vaccines, which is both clearly achievable, and really difficult (because some areas are hard to work in.)
So, it's an interesting question. But equally interesting is that no simple answer is really useful.
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