But, they appear far from being ready to say, "Let it be."
The fight over the wealth of the former Beatle, estimated at as much as $1.6 billion, could produce Britain's largest-ever divorce settlement.
Mills, 40, looked tense and said nothing as she entered a courtroom at London's Royal Courts of Justice wearing a gray suit and bright pink shirt.
McCartney, 65, arrived at court after his estranged wife, also wearing a gray pinstriped suit. He said "Good morning" as he entered court carrying a large black case.
Unlike most British court hearings, divorce proceedings are heard in private, and the door of Court 34 bore a sign reading "No Admittance - Strictly Private."
The hearing before Justice Hugh Bennett is scheduled to last five days.
The terms of any settlement would not become public record unless it moved to the Court of Appeal, or either of the parties chooses to reveal details.
That hasn't stopped a frenzy of media interest and speculation. Dozens of photographers staked out entrances to the court Monday, while two news helicopters hovered overhead.
Press reports have suggested that McCartney has offered his wife around $50 million, and that she is seeking at least double that amount.
"Current estimates suggest that Heather is likely to receive anything from 50 million to 100 million pounds ($100-200 million) as her final settlement," said Suzanne Kingston, a family law expert.
There were rumors over the weekend that the couple had reached a settlement, reports CBS News correspondent Sheila MacVicar, with Mills offering to take a much smaller sum, in exchange for not having to endure what will undoubtedly be a rough and even humiliating ride in court.
However, MacVicar adds, no such deal -- and Bennett will have his hands full.
British attorney Mark Stephens pointed out to Early Show co-anchor Maggie Rodriguez Monday that "Beatrice will get a significant chunk of change. ... (The settlement) will be divided up between, on the one hand, Heather Mills McCartney, who won't get very much at all in the overall scheme of things, and Beatrice, who will be exceptionally well provided-for with trusts for her future and such, and of course, then they'll move on to care and maintenance of the child, and make sure that she's well looked-after, between the couple."
"The fact that this has been a short marriage will be taken into account, and this could involve looking at the wealth that has been generated during the course of the marriage, as distinct from premarital wealth," Kingston added.
Mills intends to represent herself. She fired her lawyers in November after a series of interviews in which she accused McCartney of failing to protect her and their four-year-old daughter Beatrice from abuse, which she says ranges from lies and slander to death threats.
"We've had death threats, I've been close to suicide. I'm so upset about this," she said in one of her November interviews. "I've had worse press than a pedophile or a murderer, and I've done nothing but charity for 20 years."
"I am the one that is abused daily," she added.
"I have protected Paul for this long and I am trying to protect him but I am being pushed to the edge and I don't want my daughter, when she is 12, going on the Internet and reading this totally one-sided story."
Mills' left leg was amputated below the knee after a motorcycle accident in 1993, and she became active in campaigning against land mines, as patron of Adopt-A-Minefield.
The couple married in June 2002, and their daughter was born in October the following year. They announced their separation in 2006, and McCartney filed for divorce alleging "unreasonable behavior" by his wife.
The current record divorce settlement in Britain is the $94 million that businessman John Charman was ordered to pay his former wife in 2006.