I plan to apply for USA universities. Is there any difference between the number of PhD opportunities in spring and fall semesters? I mean in which semester do I have a better chance to be approved?
In my country application for spring semester is not so usual. What about USA?
Most universities offer Doctoral admissions exclusively in fall semester (once a year). Deadlines for applications are typically in December. For more detailed information, you should check the particular department's webpage.
All institutions I know of in the US have deadlines for submitting applications to the doctoral program. The earliest ones I remember were in December, and the later ones were in early spring.
Your chances of being accepted are zero if you submit after the deadline, and if you submitted early it would be at best annoying. So long story short just follow the rules of the institution to which you are applying.