How often do you use Inspiration in your D&D 5E game?
Never.
The main reason however is the mere fact that in our current family game we're not using traits/ideals/bonds/flaws, and in turn the reason for that is not wanting to introduce all parts of the game system at once to the children, so that they can learn one aspect of the game at a time. In addition, I wanted them to roleplay freely and
then figure out how to represent their PC's characteristics using the T/I/B/F system, but we haven't gotten there yet.
Still, I am not sure we will use the Inspiration rules even after we start using the T/I/B/F. There are a few things I don't like much about Inspiration (or should I dare say... I am not
inspired to use Inspiration?

), but this might be because of how I've seen it used, and since the rules are quite loosely defined, then I might just not have been lucky to see it used
well.
Certainly, one thing I don't like is how typically Inspiration is gained once and used later, it's just too gamist for me. If we eventually use Inspiration, I will require that it is used
soon, by which I don't mean
immediately but at least I want it used in the same
scene. I don't want someone to gain inspiration in a social situation in the morning and use it for saving against a trap in the evening. I want instead to keep a clear narrative connection between gaining and using it: if you get inspired while negotiating with the local lord, you earn the benefit for the negotiation itself. [Note: I don't think it is clear at all how long you can wait until using inspiration... the rules don't set any limit, but the examples make it sound like it makes sense to use it immediately, but then again the bits about not stacking multiple inspirations kind of implies you can wait]
I also don't really get the point of passing Inspiration to another player. That just doesn't make sense to me. If the other PC does something great that deserves gaining inspiration, she should gain it without someone else giving up their own, otherwise she should not gain it.
Finally, I also wish that the benefit was something more exciting than advantage. It is certainly useful, and the RAW doesn't exactly specify if you can't
first make a roll and
then decide to use inspiration if you rolled poorly, meaning you don't easily waste the benefit. But perhaps I would like a more exciting boost, such as regaining the use of an expended daily special ability, or gaining a second bonus action or reaction for the round.