When 15th c. men-at-arms freely agreed to fight on foot with swords, how often did they choose the edge and how often the point? Often we can't tell because the description of the deed is ambiguous. Also, the sword seems to have been a distinctly less popular choice for these combats than the pollaxe during this period, so examples of sword combats are somewhat limited. Still, we do have some accounts that shed light on the question.
Arms the Seneschal of Hainault Did on his Voyage to Santiago: 1402-1403. The three combats at Bourdeaux, against Alvaro d'Avile and against Rodis de Mendoza all speak of pushes of the sword, attacks with the point rather than the edge.
Continge vs. de Bars (1415) . Striking with point and edge of the sword.
Galiot de Baltasin and Phillipe de Ternant Fight with Swords, 1446 Attacks only with the point.
How Sir Jacques de Lalaing did arms in Scotland, 1449. According to Chastellain’s account, in the course of his combat Jacques drew his sword, which was an slim estoc designed only for thrusting.
Jacques de Lalaing and Jacques d'Avanchies Fight with Swords, 1450
De la Marche's Account
So the evidence of these accounts is that the edge was sometimes used, but thrusting was far more common.