申し訳ございません
Oftentimes, I hear people say this when they apologize. Recently, I’ve learned that you can also say
申し訳有りません
But I’ve never heard anyone actually say it in this form. Is this form common?
Yes, 申し訳ありません is perfectly natural. It's politer than すみません but less polite than 申し訳ございません. Hotel clerks may stick to 申し訳ございません, and you should use 申し訳ございません in a serious formal apology, too, but there are cases where 申し訳ありません is enough.
EDIT: For example, if you're saying "I'm sorry" to your close boss in the same section, 申し訳ございません can be an overkill, and I usually say すみません or 申し訳ありません. Of course this largely depends on your workplace, so if everyone around you is saying 申し訳ございません even in this situation, you may want to do the same.
申し訳ございません is more polite than 申し訳ありません, but both are very common. (In a store you might hear the former more often, since in this setting it's common for the salesperson to use the most polite form.)
申し訳ないです is also very common, but ないです is a more informal version of ありません that would be used in a more casual setting.