There are lots of good reasons to ask for a raise:
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Inflation is at 7.5%
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You like your job, but feel you are underpaid
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You don't like your job and need an incentive to stay
Lots of reasons!
A mistake a lot of people make is in assuming they can't control their pay, so they don't bother to even try. Changing jobs can be both stressful and disruptive, so your first order of business should always be to try and improve your current job.
If pay is an issue, simply meet with your boss (or HR) and tell them you want a raise. Be clear. “I'd like a 10% raise”, not “what are the chances of me getting a raise?”
If they ask why, give them a non-confrontational reason. The easiest nowadays is inflation. “Inflation is really high now and I need a raise to counteract it”. Or, if you are clearly underpaid for your job, do a little research beforehand so you can say “I did some research and the average employee with my job title and my experience is making X, but I am making Y. I would like to make X”. Or maybe something specific to your job like, “After so and so left, his responsibilities were added to mine with no increase in pay”
Unless you have an exceptionally good working relationship with your boss, you need to keep this completely non-confrontational. You should not be threatening to find a better job or anything like that. While this might succeed in pressuring them to give you a raise, you will be poisoning the well and there's a real chance they will replace you at the earliest opportunity.
If they give you a raise, great! Mission accomplished! This is a tight job market and their willingness to compensate you fairly will tell you all you need to know about how much they value you.
If they decline, it doesn't matter what their excuse is. Just quietly accept it (so you don't risk your current job), and then start looking for another job. Once you have that, just give your standard notice and ignore any subsequent attempts to retain you. They will know they had their chance and blew it. Backing out of your new job will only serve to alienate your new boss as well.
If they give you a raise but it is inadequate (or you still don't like your job enough to stay) then of course start looking for that next job. The upside is that you have a small raise in the short term and can negotiate your new salary based on your higher current salary.
The job market is very tight right now and workers have an incredible amount of leverage. However, leverage is worthless if it goes unused.