In most of the DC Continuities, (Golden Age, Silver Age, Modern Age, Pre-Crisis, Post-Crisis and the DCnU) Batman/Bruce Wayne has maintained a professional distance from all women. Dating enough to be seen to be a lax and indifferent playboy, but rarely allowing a woman into his life, permanently.
Earth-Two (Golden Age)
Alfred in DC's Super Spectacular (1975) reveals a collection of beauties which had the Dark Knight's attention.
However, in the Golden Age, Batman marries Selena Kyle (Catwoman). It is one of the only continuities (in print) where Batman chooses to actually marry and stay married to anyone.
In the 1970s comics, a series of stories taking place on Earth-Two (the parallel Earth that was retroactively declared as the home of DC's Golden Age characters) reveal that on that world, Selina reformed in the 1950s (after the events of Batman #69) and had married Bruce Wayne; soon afterward, she gave birth to the couple's only child, Helena Wayne (the Huntress).
Earth-One (Pre-Crisis)
On Earth-One before the Crisis on Infinite Earths, Batman and Catwoman were getting to be very chummy, even fighting crime together. This relationship would not be taken further due to the Universal reset.
Post-Crisis
(See StackExchange Article: Is Batman's 'relationship' with Wonder Woman comic-book canon?)
Wonder Woman. Did Batman and Wonder Woman share a more than platonic relationship? And was it canon? - Yes and yes. A brief relationship is hinted at in the DC Animated Series, Justice League Unlimited (You can see an episode featuring that relationship called "This Little Piggy.")
In the Post-Crisis Universe, they had a brief relationship in the Justice League of America comic written by Joe Kelly in The Obsidian Age Saga.
With the creation of the DCnU, any previous relationship between the Batman and Wonder Woman has been erased. Now Wonder Woman and Superman are an item but no knows how long their relationship may last.
DCnU
With the advent of the new 52 DC New Universe (DCnU), Batman has a complicated social life. Batman has child with Talia (daughter of Ras Al Ghul, supervillian mastermind) and is also dating the DCnU Catwoman. How much interaction he has with Talia is unknown but you can expect to see her again. He and Catwoman are going at it hot and heavy.
How long this lasts, where this will be going is anyone's guess. But seeing how long this relationship has lasted (through five universes, so far, not counting the dozens of imaginary tales, movies, throw away models, sex supervillainess, glimpses of the future and the like) I would count Selena Kyle (however they are spelling these days, as the woman most likely to catch, The Bat.
