Bruce Wayne commonly is referred to as a "playboy" in the comics. In the Golden Age he was linked to a number of women, and engaged to a few as well. From a statistical point of view, the most interesting feature is the steep drop off of romantic activity at the end of 1942 and beginning of 1943. After this point Batman has virtually no romantic activity for the duration of World War 2.
Between 1940 and the end of 1942 Batman was in some way romantically linked to women named Julie Madison, Linda Page, The Cat (Catwoman), The Canary (a short lived Penguin accomplice), Viola Vane, Elva Barr, and a woman only called 'Queenie'. A total of at least seven romantic partners, with two being prolonged marriage engagements that spanned multiple stories.
From 1943 until the end of World War 2, Batman was not seen as romantically linked to anyone.