Problem is the most common one. We use it in all kinds of situations. We can have big problems or small Problems.
Problem is a matter or situation regarded as unwelcome, harmful, or wrong and needing to be overcome.
A problem is often large enough to alter life either temporarily (临时的) or permanently (永久的). A problem is something bad and difficult that you have to deal with.
You can use problem in any negative (消极、负面) situation.
Problem指的是不愿看到的、不愿经历的、不容易处理的问题。暗含着事比较大，足以暂时甚至永久地改变生活。因此problem隐含着负面态度的意思。
The most common meaning of issue is the diplomatic alternative to problem. It means that we can solve things easily.
Because issue is much softer than a problem and has a readily (容易地) recognizable solution.
So especially in politics or the business world, the word problem should be avoid as much as you can. To be more diplomatic (外交的，老练的), we use the word issue as long as you want to keep a positive relationship.
Issue语气比problem缓和的多，因此在外交、商业场合通常作为problem的替代词，即便真的是problem。除非不想继续做朋友了。大家可以留意一下美国白宫对各个事务的措辞，外交交往学问很深。
Be aware, there is some overlap (重叠) between issues and problems. For example, climate change can be considered as both an issue and a problem depends on how you think about it.
Trouble can imply several related problems that are related to one thing. Trouble means that you did something wrong. A common use of this word is with the phrase “in trouble”.
For example, at the beginning, he had trouble performing in public, but now he does it well.
Trouble可用来暗示很多问题、麻烦集一身，没有头绪。用的很广的一个短语就是in trouble。
英语单词 problem, trouble以及 issue的区别
