Vox has an in-depth analysis, also discussed on the most recent episode of The Weeds podcast:

All of them cost a lot — but they all cost about the same or less than the recent round of Republican tax cuts. All five are more ambitious than any cash proposals during the 2016 primary, or any other Democratic primary I can remember. And in a head-to-head matchup of all five, two plans (surprisingly) stood out as doing the most to reduce poverty at the least cost: bills from Cory Booker and Kamala Harris subsidizing rent for low-income households.

(Emphasis mine.)

The journalist worked with researchers at Columbia to do a bunch of interesting analyses. The bills have somewhat different goals (e.g., some are targeted at fixing childhood poverty specifically) and cost different amounts. But the article looks at how many people in poverty and in deep poverty would be helped by each proposal, as well as the approximate return on investment (holding cost steady). In all analyses, subsidizing the rent of people paying over 30% of their income to rent is highly effective.

As a Bay Area resident, I think a lot about how hard it is to live here unless you’re fortunate enough to have a very high household income. And I know we’re not the only US housing market that’s tough for renters. Still, I didn’t realize that subsidizing rents nationwide would have such a high impact on US poverty.

Also, an important note: most of plans are not mutually exclusive! Several could be combined, and some of the same folks are sponsoring multiple bills.

No — it’s been well studied, and economists on the left and right seem to agree it’s counterproductive for the people it’s supposed to benefit.

According to The Economist (“Do Rent Controls Work?”):

As Paul Krugman wrote in the New York Times in 2000, rent control is “among the best-understood issues in all of economics, and—among economists, anyway—one of the least controversial”. Economists reckon a restrictive price ceiling reduces the supply of property to the market. When prices are capped, people have less incentive to fix up and rent out their basement flat, or to build rental property. Slower supply growth exacerbates the price crunch. And those landlords who do rent out their properties might not bother to maintain them, because when supply and turnover in the market are limited by rent caps, landlords have little incentive to compete to attract tenants. Rent controls also mean that landlords may also become choosier, and tenants may stay in properties longer than makes sense. And some evidence shows that those living in rent-controlled flats in New York tend to have higher median incomes than those who rent market-rate apartments. That may be because wealthier households may be in a better position to track down and secure rent-stabilised properties.

And the NY Times (“The Perverse Effects of Rent Regulation,” by Adam Davidson of Planet Money):

[T]hese programs actually make the city much less affordable for those unlucky enough not to live in a rent-regulated apartment, Mayer says. The absurdity of New York City’s housing market has become a standard part of many Econ 101 courses, because it is such a clear example of public policy that achieves the near opposite of its goals. There are, effectively, two rental markets in Manhattan. Roughly half the apartments are under rent regulation, public housing or some other government program. That leaves everyone else to compete for the half with rents determined by the market. Mayer points out that most housing programs tie government support to an apartment unit, not a person. “That is completely nuts,” he says. It creates enormous incentive for people to stay in apartments that no longer fit their needs, because they have had kids or their kids have left or their job has moved farther away. This inertia is a key factor in New York’s housing shortage. One East Village real estate agent told me that only 20 to 30 units are available in the entire area any given month.

This might be acceptable if all the rent-controlled and rent-stabilized units were inhabited by the poor people the programs were designed to help and if most poor people lived in rent-regulated units. But according to data from N.Y.U.’s Furman Center for Real Estate and Urban Policy, a majority of people in rent-regulated Manhattan apartments make far above the poverty level.