The two met in private at first, accompanied only by their translators.
Mr Trump has said he has a "good feeling" about Tuesday's much-anticipated summit.
Perhaps most importantly, we're asking Kim for something very big, something we want very much, and something he is disinclined to give.
The smiles on Tuesday "don't mean anything" unless they are followed by specific commitments to make change in North Korea, Mr. Pinkston said.
Success in Singapore would see Kim making a bold decision to exchange his nukes for economic support and security assurances, according to Ryan Haas, an Asia expert at the John L Thornton China Center.
Earlier in the day, Trump tweeted a jab at those who doubted the meeting. During his stroll, crowds yelled out Kim's name and jostled to take pictures, and the North Korean leader posed for a selfie with Singapore officials. "We have watched the struggles, and the sacrifices and the progress of the people", he said.
On Tuesday, famous "North Korean envoy" Dennis Rodman broke down in tears live on CNN, saying that the controversial talks between him and Kim Jong-un and between Donald Trump and Kim Jong-un were the right thing to do.
Tokyo's approach so far has been to talk to Pyongyang through Washington, which already has Kim's ear.
In March, Trump accepted Kim's invitation to meet, breaking with predecessors who had shunned any type of a summit without major concessions from North Korea or multilateral talks that included a host of other Asian countries.
"I truly hope it will be a successful summit that will open a new age for the two Koreas and the United States and bring us complete denuclearisation and peace".
Shortly after a group of suited North Korean diplomats set out from their Singapore hotel on Monday for talks with US officials on the eve of a historic summit, a bigger group of North Koreans headed out in summery shirts for some shopping.
In 2015, it served as the location for the meeting between Chinese President Xi Jinping and Taiwanese President Ma Ying-jeou.
It is the first time sitting leaders from both countries have ever met and marks a remarkable turnaround in the men's personal relationship.
Mr Trump has described the Singapore summit as a "one-time shot" for peace.
Still, "the friendly greeting suggests that both Kim and Trump want to have a good meeting that sets a positive tone for the hard negotiations to come", said Michael Kovrig, senior adviser for North East Asia with the International Crisis Group. "He won't have that opportunity again".