The first time Logan and Domino got together was after a mission (New X-Men Annual No. 1). During the appropriately named mini-series X-Force: Sex and Violence Domino called Logan out. She said Wolverine’s wanted her ever since their night in Japan. Later in that series, the pair began their friends-with-benefits relationship.
This relationship is ranked high because of their understanding. At no point did their relationship affect their work. When both of them went on to date other people, they never made it awkward.
Yukio and Wolverine had a similar relationship to the one Logan had with Domino. The difference is they may actually have loved each other - so much that they know it would never work between them.
Yukio is a free spirit who would never want to be tied down to one person. Logan's loyalty to the X-Men means he wouldn't be around long enough to truly be with her. So, to quote Jay-Z, their time together is their time together and their time apart is their time apart. It’s an understanding that’s worked for years.
3. Melita Garner
Don’t be upset if you don’t know her name. Melita Garner’s life with Wolverine was short-lived (it started in Wolverine Weapon X by Jason Aaron). However, she’s one of three women Logan trusted with everything he had.
Melita was in his mind fighting demons (Wolverine vs the X-Men) and one of the only women he’s brought home to meet the X-Men. And Melita was never scared of his life despite being human.