The book outlines grievances and bitterness in the Royal Family, such as a claim he and Prince William urged their father not to marry Camilla.
But one of the most striking claims from Harry, first reported by the Guardian newspaper, was how he was physically attacked by his brother.
Kensington Palace and Buckingham Palace have both said they will not comment.
BBC News has not yet seen a copy of Spare, which is due to be officially released on 10 January.
The Guardian has seen the book, while the Sun and others obtained a Spanish version after it was published in Spain ahead of its official release date. Reuters news agency has also seen the Spanish version.
Further allegations from the book have been made public since the Guardian's first article on Wednesday evening.
Among them are Harry's claim that William and Catherine, now the Prince and Princess of Wales, "howled with laughter" after seeing him dressed in a Nazi uniform for a party, the New York Post reports.
There are also accounts of Harry's drug-taking and experiences as an Army helicopter pilot in the Afghanistan conflict.
Reconciliation and compromise are not on the agenda from the leaked excerpts. So far there is a tone of unresolved grief, grievance and accusation in Harry's claims.
It is intensely personal too, about his closest family, brother, step-mother, sister-in-law, father. There is an angry cloud hanging over these claims and it is going to be impossible for that to be ignored the next time Harry is seen with the Royal Family.
The coronation is only a few months away and the run-up could become a will-he, won't-he come story about Harry.
These are some of the key claims and revelations from the book:
Harry writes that he and William begged their father not to marry Camilla, now Queen Consort, over fears she would be their wicked stepmother, the Sun has reported.
The Sun, which obtained a Spanish copy of the book after it was accidentally published ahead of its official release date, reports that Harry claims he and his brother had separate meetings with Camilla before she officially joined the family.
Harry alleges that he pondered whether she would one day be his "wicked stepmother", but that he and his brother were willing to forgive her in "their hearts" if she could make King Charles happy.
There is no detail given on when the meeting took place or how old Harry was at the time.
Harry describes how his sadness over the death of his mother, Diana, Princess of Wales, led to him seeking help from a woman who "claimed to have 'powers".
"Your mother says that you are living the life that she couldn't live," Harry says the woman told him. "You're living the life she wanted for you."
Diana was killed in a car crash in Paris in 1997 when Harry was 12.
Harry's account of a conversation he says he had with his late mother is short, the Guardian says. There is also no detail on where or when the meeting with the woman took place.
Harry claims his brother grabbed his collar, ripped his necklace and knocked him to the floor at his London cottage.
The book sets out an argument between the pair, which Harry claims was sparked by comments made by William about Meghan.
Harry writes that his brother was critical of Meghan with William describing her as "difficult", "rude" and "abrasive".
The Duke of Sussex said that his brother was "parrot[ing] the press narrative" as the confrontation escalated, according to the Guardian.
Harry is said to describe what happened next, including an alleged physical altercation.
"He set down [a glass of] water, called me another name, then came at me. It all happened so fast. So very fast.
"He grabbed me by the collar, ripping my necklace, and he knocked me to the floor.
"I landed on the dog's bowl, which cracked under my back, the pieces cutting into me. I lay there for a moment, dazed, then got to my feet and told him to get out."
The memoir also claims William "howled with laughter" when he saw his brother dressed in a Nazi uniform before a fancy dress party in 2005, the New York Post reports.
The New York Post reports Harry asked William, and his future wife Catherine, whether he should wear the costume, or dress as a pilot - and claims the pair laughed and said the Nazi uniform.
Harry was 20-years-old when the Sun published a front page photo of him dressed in the uniform at a costume party with a "Native and Colonial" theme.
Harry says he was offered a line of cocaine at someone's house when he was 17 and admits taking the drug on several other occasions, although he did not enjoy it.
He writes: "It wasn't much fun and it did not make me feel especially happy as it seemed to do to everyone else, but it did make me feel different, and that was my main objective.
"I was a 17-year-old boy ready to try anything that altered the pre-established order."
He also recounts smoking cannabis in a bathroom at Eton College while a pupil, as the Thames Valley Police officers serving as his bodyguards patrolled the exterior of the building.
While serving as a helicopter pilot in Afghanistan in 2012-13, Harry says he participated in six missions, all of which involved deaths, but saw them as justifiable.
"It wasn't a statistic that filled me with pride but nor did it leave me ashamed," he writes. "When I found myself plunged in the heat and confusion of combat I didn't think of those 25 as people. They were chess pieces removed from the board, Bad people eliminated before they could kill good people."
Harry claims the royal household dragged its feet over the date and venue for his wedding with Meghan.
He says when he consulted his brother about the possibility of marrying in Westminster Abbey or St Paul's Cathedral, William said it was not possible because they had been the venues for the weddings of Charles and Diana and William and Catherine, respectively.
William instead suggested a village chapel near their father's Highgrove House home in the Cotswolds, says Harry.
Harry and Meghan tied the knot at St George's Chapel, Windsor Castle, in May 2018.