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Browse his social media. Notice how she acts around you in social situations. Talk to mutual friends.
If you are still friends on social media, observe his postings and interactions carefully. Is he posting lots of vague and/or sad posts (sad songs about lost love, etc.)? Is he commenting on old pictures of the two of you or "liking" them? If so, it may be a sign that he is having a hard time dealing with the break up. Remember though that social media is not always an accurate depiction of what's going on in someone's life. Even someone who posts lots of pictures looking like he has the perfect life could be dealing with major emotional issues. Don't go overboard with browsing social media. Respect the privacy of your ex, and limit yourself to checking once each day, at the most. If you and your ex still hang out in a group of mutual friends, carefully (but discreetly) observe how she behaves when you are in a group of friends together. If your ex seems agitated when hanging out with you in a group, and tries to avoid interacting with you, it might be because they she is still dealing with lingering feelings. Be careful though. Your ex may still be dealing with old feelings, but that doesn't necessarily mean she is missing you. For example, she may be very angry at you because you really hurt her with your actions. Try to keep your judgements about her behavior within the context of your breakup and past interactions. Take note if your ex constantly glances at you even when interacting with other people. This could mean that she is interested in observing your behavior to see how you're feeling, too. If you have mutual friends that you trust to keep quiet about your investigating, ask them if your ex has mentioned anything about you. Your mutual friends will probably be able to give you great insight into how your ex is doing. If you have mutual friends, but are afraid they'll tell your ex you asked about him, you can try asking very casually. For example, instead of being really direct, you could say something like, "I was just wondering how [name of your ex] is doing? I know he had a big exam coming up, and I hope it went well." They may still catch on, but it won't be as obvious as saying, "Has [name of your ex] said anything about me?" Avoid constantly hassling your mutual friends about the topic, though. It's OK to bring it up once or twice, but if you bring it up all the time, they will probably get annoyed. If your mutual friends say something like, "I'm sorry, but I really don't want to get involved in the situation" then respect their wishes. This doesn't mean they don't care about you; it means that they care about both and don't want to get dragged into a "he-said-she-said" scenario or choose sides.