Attorney General J. Marshall Coleman announced yesterday that Virginia State Police have found "no criminal wrongdoing" in the handling of the Donna Shoemaker-Alan Foreman murder investigation in Arlington, and that Coleman therefore is closing the state's inquiry into the matter.

Coleman's decision is vindication for Arlington Commonwealth's Attorney William S. Burroughs Jr., who became the focus of the three-month State Police investigation.

"I'm glad it's all over," Burroughs said yesterday, adding that he was not surprised at the result.

Coleman's prepared statement dealt briefly with a report in The Washington Star last week that State Police were highly critical of Burrough's performance, had found alleged instances of ineptitude and recommended that Burroughs be replaced in the murder case with a special prosecutor.

"Contrary to published reports," the statement said, "the State Police report contains no recommendation for . . . the appointment of a special prosecutor, and offers no conclusions about the competence of any persons involved."

Burroughs said yesterday he still did not know what allegation had been made against him but he intends to find out and determine why an investigation was started.

The attorney general took the unusual step of initiating the State Police probe of the murder investigation last April 26 following camplaints by the Arlington police and a few citizens that Burroughs was failing to pursue the county police investigation of the murder.

Foreman, a 26-year-old real estate salesman, and his fiance, Shoemaker, 25, were found shot to death in Foreman's garage on May 8, 1977. Burroughs secured indictments against three persons in the crime, but later dropped charges against two of them and the third was acquitted after a two-week trial.

County police who continued the investigation became convinced that some of the original suspects were indeed involved, and before complaining to the attorney general's office, demanded that Burroughs either bring new indictments to get out of the case. Burroughs, unconvinced that police had enough evidence to indict anyone, refused to do either.

At the outset of the state investigation, a spokesman for the Attorney General said "it has been alleged that there hasn't been a followup, or that there hasn't been a prosecution when there should have been," and promised that the state's investigation would settle the matter "once and for all."

While this spokesman was careful "not to accuse anybody of anything," it became clear that Burrough's conduct in the case was the focus of the probe. Two state policemen assigned to the Culpeper office spent three months asking policemen and others what Burroughs near the end of their investigation and came back once more for followup questions.

Burroughs has maintained all along he acted properly in the case.

Informed of Coleman's decision yesterday. Arlington Police Chief Roy C. McLaren Said, "I accept that as a very fair statement." Last April, before the state investigation began, McLaren had said that if Burroughs remained on the case, "it will be a problem."

Police sources said yesterday that they were alleged Burroughs was guilty of criminal conduct or was medically incapable of the prosecution the only reasons under law that Burroughs could be replaced. For this reason, these sources said police never expected the state investigation to solve their problem.

"They (the police) think they have the case solved, but they do not," Burroughs said yesterday. "I have told them to get out and complete the investigation."

Meanwhile, according to informed sources, some police were resentful and frustrated yesterday over county manager W. Vernon Ford's decision to withdraw McLaren's request for an investigative grand jury to look into the murders.

McLaren made the request to the Circuit Court early last week and after Burroughs complained police were trying to go around him, Ford withdrew the request, saying there were other ways to get the information police wanted.

McLaren said yesterday he had asked for the special grandjury mainly to secure the testimony of people reluctant to cooperate and to place them under oath.

"What's Burroughs afraid of? Why not let an independent group of citizens evaluate the evidence?" one police source said yesterday.

Burroughs has said a special or investigative grand jury does not have any legal role in this type of murder investigation. McLaren said yesterday he will defer his request for now and do more research on "the entire subject of the special grand jury."