I will summarize the evidence for each of the imprisonments, there is more to be said than this, but this is enough to orient our thinking for now.
Rome, A.D. 60-62. The traditional view assigns the captivity Epistles to Rome. We know from Acts that Paul was in fact placed under house arrest in Rome for two years (Acts 28:30). “House arrest” means Paul was free to proclaim the gospel (Acts 28:16, 17, 23, 31; Eph 6:18-20; Phil 1:12-18; Col 4:2-4). Paul mentions the “palace guard” and the “emperor’s household” in Phil 1:13 and 4:22, implying he is in Rome. Paul greets Aristarchus in Col 4:10, in Acts 27:2 he accompanied Paul on the journey to Rome.
Ephesus, A.D. 52-55. There is no mention in Acts of any imprisonment in Ephesus, though in 2 Cor 6:5 and 11:23 Paul does say that he has often been in prison. Acts records no imprisonment until Philippi (Acts 16:19-40). Where were the others? One possibility is that these occur before Acts 13, another is that there was an imprisonment in Ephesus which is not recorded in Acts. In 1 Cor 15:32 the apostle speaks about fighting wild beasts at Ephesus. That may be a proverb or merely a metaphor. But if taken literally, it could mean that Paul was actually thrown to the lions in the arena. In 2 Cor 1:8-10 Paul alludes to some serious trouble that overtook him in the province of Asia, and in Romans 16:3, 4 he tells us that Priscilla and Aquila risked their lives to save him. We know that the pair were with Paul in Ephesus, and this opens up the possibility that it was here that they protected him. Ephesus is a natural location to send letters to the cities in the Lycus Valley. Epaphras would have the shortest route to reach Paul from Colosse (Col 4:12; Philem 23) and Epaphroditus from Philippi (Phil 2:25-30). Ephesus has a large Christian community which would assist Paul writing the letters (Col 4:10, 11). Paul asked Philemon to have a guest room ready for him in Colosse (Philemon 22) when he was released implying that he was nearby. Finally, Onesimus is more likely to have fled to Ephesus than Rome.
Caesarea, A. D. 58-60. While this appears to be the weakest possibility, Paul was in prison in Caesarea under “open arrest” for more than two years. Like Rome, he likely had enough freedom to produce short letters. He was under house arrest in Herod’s palace (Acts 24:23) and his friends were allowed free access to him. The best arguments for Caesarea require Ephesians, Colossians and Philemon to be written and delivered at the same time. The runaway slave Onesimus escaped from Colossae to Caesarea (some five hundred miles, rather than to Rome), Paul sent him back to Philemon with that letter along with Tychicus, the bearer of Ephesians and Colossians. If Ephesians was written from Caesarea, Tychicus and Onesimus would have brought the letters of Colossians and Philemon to Colossae first, then he would move on alone to Ephesus. With respect to Philippians, the distance from Caesarea to Philippi is less than to Rome, but not particularly conducive to several trips implied by the letter.
By way of conclusion, a location of Ephesus for Philippians is attractive, although the fact there is no clear reference to imprisonment in Acts or the other letters makes this a tentative suggestion at best.