Probably not until and unless the process server gets the correct address and actually serves you. Then the documents should explain the matter fully.
If the person who was attempted to be served took note of the court involved, and told you what court it was, you could call the Clerk of the Court and inquire.
Otherwise you would need to ask every possible court, which would take a great deal of time and effort.
You have not been lawfully served (at least not in most US jurisdictions) until you have been served in person, or perhaps by mail, or by publication in a newspaper, or in some other way considered lawful in your jurisdiction, but serving a person at your old address is not likely to be valid service. (Valid methods differ from one jurisdiction to another, and in some situations differ by the kind of case involved.) If the person at your old address gave the server your new address, s/he will probably be along shortly.
If a process server is given an address by the client (plaintiff), s/he may well go there first, and only do research later in case the first address is wrong. One need not worry about it until the papers are served, but it might be wise to read the legal ads in any nearby large newspapers for a few weeks, in case of service by publication. The papers should give the name of a court, and perhaps the name of a judge. You can call the clerk of the court and find out if the papers are legit. There may well be a docket no or case no or some other identifying umber, as well. This will help in verification. Docketed cases may be listed on a court web site.
A comment asks is service by publication is still possible. It can be. According to the Michigan Court Rules Rule 2.106 (D):
(D) Publication of Order; Mailing. If the court orders notice by publication, the defendant shall be notified of the action by
(1) publishing a copy of the order once each week for 3 consecutive weeks, or for such further time as the court may require, in a newspaper in the county where the defendant resides, if known, and if not, in the county where the action is pending; and (2) sending a copy of the order to the defendant at his or her last known address by registered mail, return receipt requested, before the date of the last publication. If the plaintiff does not know the present or last known address of the defendant, and cannot ascertain it after diligent inquiry, mailing a copy of the order is not required.
In addition, subrule (E) provides that:
If the court orders notice by posting, the defendant shall be notified of the action by
(1) posting a copy of the order in the courthouse and 2 or more other public places as the court may direct for 3 continuous weeks or for such further time as the court may require; and
(2) sending a copy of the order to the defendant at his or her last known address by registered mail, return receipt requested, before the last week of posting. If the plaintiff does not know the present or last known address of the defendant, and cannot ascertain it after diligent inquiry, mailing a copy of the order is not required. The moving party is responsible for arranging for the mailing and proof of mailing.
Thus if the plaintiff does not know and cannot determine the defendant's address, or has an incorrect address but thinks that it is correct, a service by publication (or even by posting) may be lawful, if the Judge so orders, without the defendant getting an individual copy of the documents by mail. This requires some unlikely events, but is possible.