Cards with Dissipate, like Whirlpool, are the most common situation where Initiative steps in. Besides that, it's actually very rare that it matters. To give you an example outside of Dissipate, we'd need to setup very specific scenarios. Such as:
During Upkeep I have a Goblin Bomber near death with a Burn condition on it, and my opponent has a Highland Unicorn that is also near death. If the Burn kills my bomber, it will immediately have a chance to trigger the Detonate attack. The attack, if triggered, has to happen at the same time that my creature dies. It's another case of two things being linked. Therefore, I probably want to roll for my Burn before the Unicorn Regenerates, so that I have a better chance of taking it with me. Obviously my opponent would want to Regenerate first. Initiative decides who must resolve their effect first.
Now, let's add an extra layer of complexity just for fun. Let's say the Highland Unicorn also has an Essence Drain enchantment attached to it. Now, the most beneficial course of events for me is to have my opponent pay the Upkeep cost, then roll for my burn, and only let them regenerate AFTER we see if my bomber explodes. However, even if I have Initiative, I can't dictate this course of events. If I let them resolve an effect before I roll for my Burn, they still get to choose which of their effects they resolve first. So, they'd probably choose to Regenerate first.
A trick that I find useful for thinking about these situations is that Initiative gets to decide who's turn it is to resolve an effect, but not which effect they resolve. This is because most, if not all, effects are clear about who controls the timing and the only reason to ever have Initiative step in is to determine the relative order between multiple effects with different controllers.
But if you aren't sure who controls the timing of an effect, then I would have Initiative decide for that as well. I can't currently think of any effects that I would say don't have a single person in control already, but there are certainly cards which aren't the clearest on who that person is. So, if it's not clear, have Initiative decide and then after the game check the Supplement or ask the forums to see if other's can help figure it out.