Inspiration as presented in 5e is kind of janky, both in the way you reward it and in the way it is used. The concept is a good idea, but its execution is hard to pull off and arguably has a very forgettable effect. The suggested way to award according to the Player’s Handbook is to give it out when someone performs an act of roleplaying one of their character’s traits – especially the flaws. A good idea, but impossible to keep track of with even a few players. You have to know what their traits are (four per player) and remember to award it for playing to their character. Depending on their traits it may come up often or not at all during a session. Not fun.
Even if you do give out inspiration often, all it does is give advantage on a single attack, check, or save. Technically you have to declare you use it before you roll and you can only give someone else inspiration for a roleplaying moment, but all of the time it’s forgotten that you have it. What normally happens then is that someone obviously fails a save, gets depressed, and then they or someone else announces: “wait, I have inspiration”. It feels bad for the DM to say you can’t use your inspiration, because it’s going to make the player grumble and then it’s just going to happen again because everyone forgets they have it yet again. So, it ends up in them getting another chance at the roll for no discernible in-game reason. I think that this sucks for everyone because you’ve just essentially had to retcon something in the game and interrupted the flow. It’s not even a cool character or class trait, it’s just an unexpected time reversal. It breaks the scene so much to have to do something so meta-game-y.
So I’ve been experimenting lately with making inspiration just a simple d10 to the roll. It seems to be working well. It is remembered quite a bit more, it actually helps if you already have advantage, and it isn’t “wasted” if you just happen to roll low. Win-win. I’ve also decided that everyone gets inspiration once the reach certain milestones in the campaign or every 2 sessions, whichever comes first (normally the milestone). That seems to work out well too. I’m likely going to experiment further and not give them inspiration, but mark down that they have inspiration myself and offer its usage if they fail something by 10 or less. I remember it much more than the players.