Whether you're an employer who wants to know whether you can trust an employee with a task or someone who simply someone who wants to confide in a friend; you'll want to know whether you can truly trust them or not.
1. Observe their behaviour. Look at the way that they act. How do they treat others? Are they inconsiderate and/or rude? Are they quick to join in on gossip? Are they quick to judge people? Do they continuously concern themselves with what other people are doing? There's a difference between someone who wants to helpful and someone who's just being nosy and wants something to gossip about but that alone can't determine whether someone is trustworthy or not. Someone that seems to be involved in a lot of drama is someone that you need to be careful about - there's a reason why they don't get along with people very often.
2. Listen to them. Communication is important. When they talk to you, are they most likely to talk about other people - and if they do, are they saying negative things about them? If this person is telling you things about other people's private lives for the sake of having an "interesting conversation", you should take into consideration that this person isn't very trustworthy. It doesn't matter whether they mention their name or not; someone who talks about other people's business (with strange intentions) after they have personally been confided in is someone that you should be wary of and you should begin to question whether you can trust them or not. If they feel comfortable with talking to you about their own personal business (e.g. things that are bothering them, asking for guidance/advice etc.), it could a sign that they somewhat trust you, perhaps take a chance and trust them too.
3. Reliability; can they keep promises? Everyone lets someone down from time to time but if someone is continuously letting you (or anyone else) down at the most important times with little-to-no excuses, then they aren't very reliable. Part of having trust in someone derives from being able to have faith in them and rely on them no matter what your situation may be.
4. Honesty. Someone who constantly lies is probably doing so as a result of trying to save themselves. Once again, intentions are another thing to consider but someone who constantly lies about themselves to make themselves look good may have something to hide. If they can lie about their own life, then imagine what else they are willing to lie about. Someone who has to lie to get themselves out of a situation isn't trustworthy, just like someone who maliciously lies about other people is spiteful.
5. Personality. Other things to consider are whether they are a self-centered person or not. Self-centered people only care about themselves and they're more likely to let you down on more than one occasion than anyone else. Some people make exceptions on the very few people that they will care about but those that never think about anyone except for themselves is someone that has the potential to betray you if you give them the chance to.
Don't file someone under "untrustworthy" because they have let you down once or twice before in the past. Learn not to hold on to the past too much because it can have an effect on whether you can trust people or not in the future (see 'Warnings').
Just take a chance on people. You'll never know if you don't find out for yourself. Start off by telling them small things, and if it gets around, then you know that you probably can't trust them.