One of the things that has stood out for me that Anthony De Mello said is that you can't really practise non-attachment until you've experienced attachment. It's easy to practise being non-attached to something you're not really attached to in the first place. It becomes much more of a practice if you let go or begin to let go of something that's actually quite dear to you without knowing if you'll get to have or see it again or not.
In the case of a death, you know that you won't get to have it again and in some ways that makes it easier. It's so final. In the case of something else, it can be difficult because you don't know if you'll have it again. So letting go in the face of uncertainty becomes that much more challenging. It could be letting go of a habit, like drinking coffee, or something small. Or it could be bigger, like letting go of a child who's going to go off and experience the world. It could be even bigger and more attachment-like if it's someone you don't want to let go but they're going anyhow, like a good friend or lover you don't want to say good-bye to.
I'm currently in the middle of practising some serious non-attachment. I'm trying to keep my arms by my sides while my whole being is screaming to grab on and hold tight.