Lacus' relationship with Athrun is probably the longest one out of all of Lacus' relationships. Their parents agreed on an arrange marriaged which made Athrun Lacus' fiancee. Apparently Athrun isn't very creative at giving different presents to Lacus, which is probably what Lacus likes about him. Their bond of understanding of each other is unquestionably strong and possibly, Athrun did truly fall in love with Lacus (who later left him to be with Kira ^^;)
Lacus and Cagalli
There isn't much between Cagalli and Lacus. There good friends but throughout the series rarely do you see them together and socializing. Their strongest times shown was in the Valley of the Stars.
Lacus and Kira
It's probably the most obvious one. Lacus first fell in love with Kira and Kira slowly followed, but he still love Fllay even to the end. Kira's heart was splitted and Lacus didn't mind because she just wants to be with him. She fell in love with his gentle heart that mades him understand that coordinators aren't all bad.
Lacus and her Father, Seigel Clyne
Lacus' father, Siegel Clyne, is the chairman of PLANT and has the traits of a pacifist which Lacus inherits. Even though her father is always at work and barely at home, she still love him dearly. They are like true family.