By "the Lord’s Supper" we are referring to the eating of simple bread and drinking of fruit of the vine (wine or grape juice) by a group of believers gathered in Jesus' name, in memory of his sacrificial death, as per Jesus' example and instruction.

Matt 26:26-28 26 While they were eating, Jesus took bread, gave thanks and broke it, and gave it to his disciples, saying, "Take and eat; this is my body."
27 Then he took the cup, gave thanks and offered it to them, saying, "Drink from it, all of you. 28 This is my blood of the covenant, which is poured out for many for the forgiveness of sins."
NIV

The first instance of the Supper was toward the end of the meal on the night of Jesus' betrayal, as reported in Matthew, Mark, Luke, and first Corinthians (John mentions the evening meal, but doesn’t specifically describe the bread and fruit of the vine). At that time Jesus described the bread as his body, and the fruit of the vine as his blood, which he presented to the disciples in the context of a new covenant. Luke tells us that he instructed them, "do this in remembrance of me." (Luke 22:19). There is clearly an instruction then for the followers of Jesus to commemorate his life and death in this ceremonial way. In that first instance reported in the gospels, the "institution" of the communion, the timing was in the evening, either Wednesday or Thursday. The day of the week is not mentioned in the text, but was followed by the day of crucifixion, and then a high Sabbath. So at the time of the institution, in the gospels and 1 Corinthians, Jesus did command his disciples to do this, but we have no record of him defining a time or frequency that evening.

In the book of Acts, which reports the founding of the church and its growth in the first generation after the cross, we have two particular mentions of the disciples eating the meal Jesus prescribed. The first mention is in the context of the new disciples after Pentecost, when 3000 were baptized, and we are told that "They devoted themselves to the apostles' teaching and to the fellowship, to the breaking of bread and to prayer." (Acts 2:42 NIV). We know that "the breaking of bread" was a matter of devotion for the Christians led by the apostles, which at least suggests some level of frequency more than annually or semi-annually, but Luke mentions no time frame in the Jerusalem church context. He does mention frequent meetings and sharing at all levels, including daily meals in the verses that follow. It may be worth noting that in verse 42 Luke reports "the breaking of bread" as a matter of devotion, with the definite article used with each of the four activities in that list. He also mentions "they broke bread in their homes" in verse 46, without the definite article, a common way of describing a regular meal. Since "the breaking of bread" was a matter of devotion, along with fellowship, prayer, and the apostles' teaching, we can surmise that the Jerusalem church must have commemorated the Lord's Supper with frequent regularity.

The next time Luke mentions the Supper in Acts is in 20:7.
On the first day of the week we came together to break bread. Paul spoke to the people and, because he intended to leave the next day, kept on talking until midnight.
Before commenting on the timing Luke specifies, the context is worthy of noting. Paul had left Ephesus on a journey toward Jerusalem, by way of Macedonia and Achaia (Acts 20:1-3). During this journey Paul wrote the letters of 2 Corinthians and Romans, and mentions the journey and the reason for it, to carry a financial gift from the gentile churches to the needy brethren in Jerusalem. He had a definite time table in mind, and faced numerous challenges in meeting that plan. Acts 20:16 tells us Paul was in a hurry, wanting to reach Jerusalem, if possible, by Pentecost and he made several adjustments to his travel plans to make that feasible. Within the context of that urgency, Luke tells us that Paul and company left Philippi for Troas after the feast of unleavened bread (or Passover, Acts 20:6). That meant that Paul had less than 7 weeks to reach Jerusalem from Philippi. However, when he arrived in Troas 5 days later (now about 6 weeks left to reach Jerusalem), rather than hurry on, Paul waited 7 days (Acts 20:6) until the first day of the week. They arrived in Troas on Monday, and waited until Sunday, intending "to leave on the morrow," which he did, walking to Assos (20:13). Why did Paul spend a week in Troas, and skip visiting other churches (20:16-17), when he was in a hurry to reach Jerusalem? The only answer Luke gives us is that the church met to break bread on the first day of the week, and that's when Paul talked to the assembled church. It is difficult to find any other rationale for Paul's travel choices than that he specifically waited a week in Troas in order to be with the assembled church when they came together on the first day of the week to break bread. The example is all the more compelling when we consider that there is never any other time frame mentioned for the church gathering to break bread than the first day of the week, that this is the explicit reason Luke mentions for the gathering on the first day, and that this is consistent with everything we know from other scriptures and from ancient history. Paul could expect the church in Troas to meet on the first day of the week to break bread.

When Paul discussed the Lord's Supper in 1 Corinthians, he emphasized several important features of its meaning in 10:14-22, and discussed the circumstances expected by the Lord in 11:17ff. Among the points Paul maked regarding the Supper in 11:17ff, is the fact that the church assembles "as a church" for the purpose of eating the commemorative meal (vv 17, 33). This is consistent with the stated purpose of the first day gathering cited by Luke in Troas. 1 Corinthians further affirms that this assembly occurred on the first day of each week when Paul directed the Corinthian church, like the Galatian church, to "lay by in store" on the first day of each week.
1 Cor 16:1-2 Now about the collection for God's people: Do what I told the Galatian churches to do. 2 On the first day of every week, each one of you should set aside a sum of money in keeping with his income, saving it up, so that when I come no collections will have to be made.
NIV

Paul's instruction about the collection by the churches makes sense only if the regular assembly of the church was already on the first day of the week.

Affirmation of the regular celebration of the Supper can be further strengthened by consideration of historical references in the late first and second centuries, with universal identification of "the Lord’s Day" (Revelation 1:10) as the first day of the week in those historical sources. Ancient writers considered the identification of the church with the first day of the week symbolically appropriate (and so do I) on the basis of it being the day of beginning and creation of light; on the basis of it being the day of resurrection; on the basis of most of the reported post resurrection appearances of Jesus in the gospels being on the first day of the week (both the day of resurrection, and the "eighth day" following, the next Sunday, the day after the seventh day); and the fact that the first proclamation of the gospel, resulting in 3000 being saved, occurred on Pentecost, the day after the seventh Sabbath after Passover (in other words, the first day of the week). God's interest in a weekly reminder for his people to trust him was certainly affirmed in the Sabbath law of the old covenant.

These are some of the reasons we have a weekly observance of the Lord’s supper, on the first day of each week. I find it compelling that this is the only New Testament example we have. Any other choice, monthly, semiannually, or whatever, would be simply arbitrary, with no precedent or Biblical imperative. Examples are in fact very important in understanding what the Lord’s will is. It seems likely from the sources we have that, most particularly the New Testament itself, the practice of Sunday observance of the Lord’s Supper was so completely pervasive and completely accepted in the apostolic period that no specific instruction to do so was needed in the corrective and instructive letters, and so the examples of Troas and Corinth are normative and sufficient.