What part of the Old Testament is applicable to Gentiles?

Most of the Old Testament seems directed to the nation of Israel. What part of the Old Testament is applicable for Gentile believers?

First, the Gentiles had little or no relationship to God in the years prior to Jesus' incarnation, as Paul explains:

Eph. 2:11 Therefore remember that formerly you, the Gentiles in the flesh, who are called “Uncircumcision” by the so-called “Circumcision,” which is performed in the flesh by human hands —
Eph. 2:12 remember that you were at that time separate from Christ, excluded from the commonwealth of Israel, and strangers to the covenants of promise, having no hope and without God in the world.
Eph. 2:13 But now in Christ Jesus you who formerly were far off have been brought near by the blood of Christ.
Eph. 2:14 For He Himself is our peace, who made both groups into one and broke down the barrier of the dividing wall,
Eph. 2:15 by abolishing in His flesh the enmity, which is the Law of commandments contained in ordinances, so that in Himself He might make the two into one new man, thus establishing peace,
Eph. 2:16 and might reconcile them both in one body to God through the cross, by it having put to death the enmity.

Prior to the advent of the Church age at Pentecost, the Gentile world was largely excluded from the grace of God. While a few Gentiles were offered opportunity to enter into the assembly of God’s people in Israel (i.e., Ruth, Rahab, etc.), the vast majority of Gentiles were “excluded” from the commonwealth of Israel.

Once Christ appeared, the New Covenant was inaugurated in His blood, and all believers (Jew and Gentile) are united in one body by faith during the Church age.

Nevertheless, all who have been united by God’s grace in the New Covenant benefit from the entirety of God’s word, both Old and New testament. All of God’s word is beneficial to our spiritual growth and relationship with the Lord. As Paul says:

2Tim. 3:16 All Scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness;
2Tim. 3:17 so that the man of God may be adequate, equipped for every good work.

While many commands found in Old Testament scripture will not apply directly to Gentiles (i.e., the Law of Moses), nevertheless study of these scriptures still benefits a Gentile believer in the New Covenant. Among other benefits, we learn about the work of God in previous ages and with other peoples, we learn about promises made concerning future events, we learn God’s standards for holiness, and we learn about Christ, Who is pictured in these things.