From the very beginning of when I started to talk and write about refactoring people have asked me how it should be incorporated into the wider software development process. Should there be refactoring phases in the software development lifecycle, what proportion of an iteration should be devoted to refactoring tasks, how should we figure out who should be assigned to refactoring duties? Although there are places for some scheduled refactoring efforts, I prefer to encourage refactoring as an opportunistic activity, done whenever and wherever code needs to cleaned up - by whoever.
What this means is that at any time someone sees some code that isn't as clear as it should be, they should take the opportunity to fix it right there and then - or at least within a few minutes. This opportunistic refactoring is referred to by Uncle Bob as following the boy-scout rule - always leave the code behind in a better state than you found it. If everyone on the team is doing this, they make small regular contributions to code base health every day.
This opportunity can come at various parts of implementing some new functionality or fixing a bug. One is a preparatory refactoring, where before you begin to implement something you see that this task would be easier if an existing class's API was structured differently. You first refactor it to how it ought to be and then start adding your functionality.
As you add the functionality, you realize that some code you're adding contains some duplication with some existing code, so you need to refactor the existing code to clean things up. This continuous attention to the code is important - but do remember that you should only refactor when your tests are green.
You may get something working, but realize that it would be better if the interaction with existing classes was changed. Take that opportunity to do that before you consider yourself done.
Sometimes you see an opportunity when you're in the middle of something else. Rather than interrupt your current thought it's useful to make a note of it and come back to it when you are ready. Don't leave it for long, come back the same day, before you've hit that final point of being done.
Some people object to such refactoring as taking time away from working on a valuable feature. But the whole point of refactoring is that it makes the code base easier to work with, thus allowing the team to add value more quickly. If you don't spend time on taking your opportunities to refactor, then the code base gradually degrades and you're faced with slower progress and difficult conversations with sponsors about refactoring iterations.
There is a genuine danger of going down a rabbit hole here, as you fix one thing you spot another, and another, and before long you're deep in yak hair. Skillful opportunistic refactoring requires good judgement, where you decide when to call it a day. You want to leave the code better than you found it, but it can also wait for another visit to make it the way you'd really like to see it. If you always make things a little better, then repeated applications will make a big impact that's focused on the areas that are frequently visited - which are exactly the areas where clean code is most valuable. Like most aspects of programming this decision requires thoughtfulness.
One of the features of opportunistic refactoring is that it can hit any part of the code base you're working in. You may be doing most of your work in one class, but spot problems in a class that's in a quite different area of the code. That lack of locality shouldn't stop you from making the change now. There's often a temptation to leave a change in another part of a code base to another day - but another day often doesn't come.
Refactoring does depend on having a good regression suite and it's wise to be wary if you think you're about to touch part of an application that's weaker on its tests than it should be. In this case remember that it's quite reasonable to throw in an extra test or two if you can do that without straying too far down the rabbit hole. I also find that making a deliberate error to see if a test catches it can be a way to get a feel for how good your safety net is.
I'm wary of any development practices that cause friction for opportunistic refactoring such as strong CodeOwnership or using a FeatureBranch. This is actually my primary concern with using feature branching . Often when people are working with feature branches, they are discouraged from opportunistic refactoring because it makes merges more difficult  - particularly if the branches live longer than a couple of days.
My sense is that most teams don't do enough refactoring, so it's important to pay attention to anything that is discouraging people from doing it. To help flush this out be aware of any time you feel discouraged from doing a small refactoring, one that you're sure will only take a minute or two. Any such barrier is a smell that should prompt a conversation. So make a note of the discouragement and bring it up with the team. At the very least it should be discussed during your next retrospective.
From the beginning I've always seen refactoring as something you do continuously, as regular and indivisible a part of programming as typing if statements. Yet there's a common misconception about refactoring in that it's something that needs to be planned out. Certainly there is a place for planned efforts at refactoring, even setting aside a day or two to attack a gnarly lump of code that's been getting in everyone's way for a few months. But a team that's using refactoring well should hardly ever need to plan refactoring, instead seeing refactoring as a constant stream of small adjustments that keep the project on the happy curve of the DesignStaminaHypothesis
My infodeck on Workflows of Refactoring talks about different ways you can use incorporate refactoring into your work.
Ron Jeffries came up with a lovely visualization to describe gradually refactoring through messy code and why you shouldn't have refactoring tasks on your backlog.
The metaphor of TechnicalDebt fits in very well with these issues.
1: Modern tooling helps, but still gets tripped by SemanticConflicts.