Is it better to apply to doctoral program in the fall semester or spring semester? 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.