Correction, second best looking. And judging by your initial post.. looks like the competition (me) has quite the lead on the runner-up (you). But I'm a generous fellow, my friend, and shall help you with a few things to help you answer your question..
Rule #1.
Stop caring about whether or not she's interested in you. You should be fun, spontaneous and unpredictable. You're the wild card. Some days, you engage in a discussion, some days, a quick chat, and others, nothing at all. There should be no routine. None, whatsoever.
Rule #2.
You start the engagements. If a girl approaches you, it's cute. You don't want cute! You engage her. If she engages back, done deal. This is as simple as raising your eyebrows and smiling when she looks at you. If she looks away, too bad, try again later. If she looks down at the ground, then back at you, that's submission, an indicator of tension, and that's good. And if she approaches to talk to you, no matter what it's about (from photocopies to her new outfit to movie reviews), again, done deal.
Rule #3.
When you talk to her, don't talk too fast. But don't talk too slowly either. Actually, just get her to do most of the talking. And when you do say something, just feed back what she said. More importantly, mirror her dialogue. Some women use words that relate to the senses.. sometimes hearing: ie "sounds" good, I "hear" you. Others use sight: I "see" what you mean, "look" at it this way. Others still use words that relate to cognitive senses: "think" about it, I "believe" that..
Find out what kind of words she uses, and then use them. It's harder than it seems/sounds/looks (pun intended). Two things can happen: A) absolutely nothing at all or B) increased rapport between the two of you.
Now.. all you have to do is look for signs..
Sign 1 Physical maneurisms of sorts. When she talks to you, does she play with her hair? Bite her nails? Play with her hands? Fingers? Does she adjust her outfit? Rub her arms? She craves physical contact.
Sign 2 Dodgy eyes, breaks eye contact. She looks down when she talks to you, looks at you, your outfit, then at you again. Her facial expressions keep changing, from a grin to a frown, to neutral to a smile, every few seconds. She blinks alot. Those are all good indicators. If she looks up or to the sides, or maintains a neutral face, she's probably not that interested.
Sign 3
Just ask her. But don't ask her. Say something like "you look like you have something on your mind"... or when she's talking and there's a pause, change the subject. If she reverts to the old topic, then too bad. But if she goes along with the new one (hell it could be about anything), she's just there to talk to you. Whatever you do, don't talk about yourself. Get her to talk about herself. For example, if you're talking about a new client account or someting, change the discussion to her shoes, or her commute that morning, or whatever.
Sign 4..
Hey now, can't give out all 'dem secrets! Hehe, j/k. TS is probably the best way to chat, seeing as I have lots to share. 
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