Polycarp's references to Paul are in a thread of mine:
viewtopic.php?f=3&t=3285.
Clement has the following references to Paul:
1 Clement 5.1-7: 1 But to stop giving ancient examples, let us come to those who became athletic contenders in quite recent times. We should consider the noble examples of our own generation. 2 Because of jealousy and envy the greatest and most upright pillars were persecuted, and they struggled in the contest even to death. 3 We should set before our eyes the good apostles. 4 There is Peter, who because of unjust jealousy bore up under hardships not just once or twice, but many times; and having thus borne his witness he went to the place of glory that he deserved. 5 Because of jealousy and strife Paul pointed the way to the prize for endurance. 6 Seven times he bore chains; he was sent into exile and stoned; he served as a herald in both the East and the West; and he received the noble reputation for his faith. 7 He taught righteousness to the whole world, and came to the limits of the West, bearing his witness before the rulers. And so he was set free from this world and transported up to the holy place, having become the greatest example of endurance.
1 Clement 47.1-7: 1 Take up the epistle of that blessed apostle, Paul. 2 What did he write to you at first, at the beginning of his proclamation of the gospel? 3 To be sure, he sent you a letter in the Spirit concerning himself and Cephas and Apollos, since you were even then engaged in partisanship. 4 But that partisanship involved you in a relatively minor sin, for you were partisan towards reputable apostles and a man approved by them. 5 But now consider who has corrupted you and diminished the respect you had because of your esteemed love of others. 6 It is shameful, loved ones, exceedingly shameful and unworthy of your conduct in Christ, that the most secure and ancient church of the Corinthians is reported to have created a faction against its presbyters, at the instigation of one or two persons. 7 And this report has reached not only us but even those who stand opposed to us, so that blasphemies have been uttered against the Lord s name because of your foolishness; and you are exposing yourselves to danger.
Ehrman writes as follows about 2 Clement on page 156 of his first Loeb volume of the Apostolic Fathers:
He also appears to allude to the writings of Paul (e.g., in the image of the athlete in ch. 7 [cf. 1 Cor 9:24-27] and of the potter and clay in ch. 8 [cf. Rom 9:19-24]), but does not cite any of his books explicitly.
Ignatius (middle recension) has the following:
Ignatius to the Ephesians 12.1-2: 1 I know who I am and to whom I am writing. I am condemned, you have been shown mercy; I am in danger, you are secure. 2 You are a passageway for those slain for God; you are fellow initiates with Paul, the holy one who received a testimony and proved worthy of all fortune. When I attain to God, may I be found in his footsteps, this one who mentions you in every epistle in Christ Jesus.
Ignatius to the Romans 4.3: 3 I am not enjoining you as Peter and Paul did. They were apostles, I am condemned; they were free, until now I have been a slave. But if I suffer, I will become a freed person who belongs to Jesus Christ, and I will rise up, free, in him. In the meantime I am learning to desire nothing while in chains.
I have read arguments (I forget where) that the epistle to Diognetus depends upon several Pauline epistles, as well.
The dates of all of these texts are pretty open to argumentation.
Ben.