Corbyn said he would look into the idea at the launch of his street harassment policy document, noting growing concern about sexual harassment on public transport.
"Some women have raised with me that a solution to the rise in assault and harassment on public transport could be to introduce women-only carriages. My intention would be to make public transport safer for everyone from the train platform, to the bus stop, on the mode of transport itself," he said.
However Cooper, said women should not have to "shut ourselves away from men for our own safety" and warned "segregation" would not work.
Burnham said Britain should not "even be considering the idea of segregated train travel" in "this day and age".
"As a Dad of two young girls, I want to see a proper society-wide strategy on tackling violence against women. We need sufficiently funded police forces, especially the British Transport Police, and investment in practical measures like better lighting to ensure safety at all times," he said.
Kendall said "gender segregation" was not the answer. "That’d be an admission of defeat, rather than a sustainable solution," she said.
However Labour MP Diane Abbott, who is seeking Labour's nomination for London mayor and is a strong supporter of Corbyn, came to his defence.
In 2014, Tory transport minister Claire Perry also said she was looking into women-only carriages on trains. However this morning education secretary and women's minister Nicky Morgan said she was "very uncomfortable with the idea".
"Women should feel safe and be free from harassment on public transport. [This idea] seems to say 'let’s segregate people' rather than tackling the issue. And so I don’t think this is the right way to go,” she told Sky News.
Cooper said in a statement on Wednesday morning: "The staff needed to enforce the segregated carriages should be keeping all the carriages safe instead. Transport bosses and police need to do far more to crack down on harassment and abuse - and that means we need more visible police and staff on tubes, trains and platforms, better lighting and CCTV, tough action to follow up abuse (which too often doesn't happen), and a complete change in culture on buses, railways and tubes.
"We need national standards and obligations for police and local councils to tackle violence against women and harassment, a new Commissioner to ensure standards are enforced, and stronger laws in place. And most important we need compulsory sex and relationship education in schools to challenge attitudes and insist on zero tolerance of violence and harassment in all its forms. All this should be part of a new Violence Against Women and Girls Bill that I want to bring in.
"The responsibility and pressure shouldn't be on women to hide away, it should be on perpetrators to halt the harassment, transport bosses and the police to tackle women's safety and everyone else to stand up against abuse."
Corbyn said that he would “consult with women and open it up to hear their views on whether women-only carriages would be welcome,” adding that he would find out if “piloting this at times and on modes of transport where harassment is reported most frequently would be of interest."
"The excellent work of individuals, campaigns, and groups like Everyday Sexism and Stop Street Harassment has highlighted just how prevalent street harassment is in our country today; and the extent to which many women feel uncomfortable, anxious, and unsafe just going about their daily routines.”
"It is simply unacceptable that many women and girls adapt their daily lives in order to avoid being harassed on the street, public transport and in other public places from the park to the supermarket,” he said, adding: “This could include taking longer routes to work, having self-imposed curfews, or avoiding certain means of transport."
Tory MP Sarah Wollaston, the chair of the health committee, also immediately criticised Corbyn's plan, tweeting: "Segregating women on public transport doesn't protect anyone, it just normalises unacceptable attitudes".
Women-only carriages are currently in service in Japan, India and Brazil.