R124 Successor, you mean.
Most of the recent first ladies are generally civil to one another, or even friends. After all, who else on the planet understands what it means to marry some schmuck, and decades later find yourself a world figure because of HIS career ambitions? It's a screwy position.
But some first ladies who didn't get along:
Mamie Eisenhower and Jacqueline Kennedy - Jackie had given birth to John-John a short time before she was due to tour the White House, and a bitter Mamie led her on a room-by-room slog of the entire mansion. Mamie knew this would exhaust her. Jackie finished the tour without complaint, and promptly collapsed at her hotel. Mamie smirked and said "She never asked for a wheelchair!"
Caroline Harrison and Frances Cleveland - Harrison was the elder figure. Her husband defeated President Cleveland in a bitter election, but Cleveland stuck around as a national figure with his comely young wife, which ate into the Harrisons' social life. Caroline looked down on Frances - who was nearly 30 years younger than Grover Cleveland and had known him since childhood. It was a creepy marriage, scandalous, but also oddly endearing to the public. Frances Cleveland was young, beautiful and demure, while Caroline Harrison was older and boring. Caroline died of TB two weeks before the 1892 election (one of three first ladies to die in office), and she didn't live to see her husband defeated by good old Mr. Cleveland.
Margaret Wilson and Edith Galt Wilson - Margaret was the eldest daughter of Woodrow Wilson, and served as his first lady during the year between the death of the first Mrs. Wilson and his remarriage to the second Mrs. Wilson. Margaret was a grown woman who loved her mother and had a frosty relationship with Edith, who was herself a headstrong old widow with an odd ability to enrapture the famously intellectual President Wilson. All three of Wilson's daughters appreciated the comfort that this new wife brought to their lonely father, but tensions remained. No record exists of Margaret's feelings about Edith Wilson during the final year of her father's presidency - when Wilson was bedridden with severe strokes, and Edith Wilson effectively served as acting president - but she was a sharp cookie who surely had her opinions.
Other first ladies got along great. Betty Ford and Rosalynn Carter both campaigned for the ERA, and Michelle Obama and Laura Bush have a good personal chemistry. Nellie Taft and Edith Roosevelt were like long-lost sisters, and got along great. Both were brassy, outspoken broads who ran their own offices, smoked cigarettes, and made political moves separate from their husbands.