Critics of the summit leapt at the handshake and the moonlight stroll Kim took Monday night along the glittering Singapore waterfront, saying it was further evidence that Trump was helping legitimize Kim on the world stage as an equal of the USA president.
The two men are expected to meet on their own for the better part of an hour, with only a pair of interpreters in the room.
Trump said over the weekend that he would know if Kim is serious about brokering peace within "a minute" of meeting him.
It says the sides "commit to establish new U.S. -DPRK relations in accordance with the desire of the peoples of the two countries for peace and prosperity".
"It's incredible, when I said those things, when I said those damn things, when I went back home I got so many death threats", a tearful Rodman told CNN's Chris Cuomo while wearing a "Make America Great Again" hat.
Trump is sounding optimistic about his ability to persuade North Korea to abandon its nuclear weapons program after a lengthy one-on-one meeting with leader Kim Jong Un.
"It was not an easy journey", Kim said, speaking through a translator.
The eccentric former basketball player and reality television star is one of the few westerners to have spent time with Mr Kim during multiple visits to Pyongyang.More news: Vettel ends Ferrari's 17-year wait for Canada pole
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On Saturday, Singaporean Foreign Minister Vivian Balakrishnan told reporters at Beijing Capital International Airport that his country would give the same warm welcome to the North Koreans as they have done to other state guests.
CNN has not seen the document, but Trump said it would be shared later that day.
For Mr Trump, he's likely to credit any success in these talks to his "maximum pressure campaign".
Kim has rejected calls to unilaterally give up his weapons in return for economic aid, and instead has proposed a step-by-step denuclearization process.
The former Central Intelligence Agency chief said USA sanctions will remain in place until Washington verifies the North's denuclearization efforts.
He added there was no mention of verifiable or irreversible denuclearization in the Singapore summit agreement.
But in return, Washington would offer "different and unique" guarantees "to provide them sufficient certainty that they can be comfortable that denuclearisation is not something that ends badly for them".
Before signing what Trump described as a "comprehensive letter", Kim said the two leaders had a historic meeting "and made a decision to leave the past behind".
But a war would mean devastating casualties, including of American troops in South Korea, and American cities would also be at risk of suffering a nuclear strike because of the North's recent advances.