DALLAS (AP) _ Griffin Street was quiet this Saturday morning, the chalk that outlined Officer John Chase's body rubbed away by a week's worth of passing feet. Chase lies in a snow-dusted Iowa grave, his mentally ill assailant in a Dallas cemetery.

But the city is still wrestling with public outcries, racial tensions, a beleaguered police department and the horror that spectators could actually cheer an officer's killing.

''It showed how vulnerable we all are, how quickly things can get out of hand,'' said Sgt. Gary Beck, Chase's supervisor.

Carl Williams, a black homeless person who had been in and out of mental institutions, snatched the white officer's gun while Chase was checking a motorist he stopped on the morning of Jan. 23. While two or three young black people at a bus stop reportedly egged him on, Williams fired repeatedly at point-blank range into Chase's face. Williams was himself killed in a shootout with pursuing policemen.

In the week since then, the case has brought heightened attention to long- running controversies over police pay, staff shortages and minority tensions, as well as Texas' well-publicized jail overcrowding and a lack of facilities for the homeless and mentally ill.

''The manner in which the officer was shot, the tragic nature of it, coupled with the issues that have been festering ... the death of the officer brought those issues to the breaking point,'' said Lawrence Redlinger of the University of Texas' Center for Applied Research, which recently did a study on race relations in Dallas.

Emotions flared when Police Chief Billy Prince, stung by a city-backed congressional hearing last summer on the department's use of deadly force, went on television hours after the shooting to blame the City Council for creating a public atmosphere so hostile that the officer's death was virtually inevitable.

All week, police investigated conflicting reports over whether anybody in the crowd actually yelled ''Shoot him, shoot him,'' as Prince claimed.

Dallas Times Herald columnist Laura Miller, quoting two witnesses who said they did not hear anyone say that, went so far Friday as to label Prince a ''fibber'' and say Dallas could not breed ''a jeering crowd of cop killers.''

But late Friday, police Capt. John Holt released partial statements from four witnesses who said they heard at least two and perhaps three young blacks in a crowd of about 15 to 20 urge the vagrant to fire while Chase pleaded for his life.

Two witnesses said a black woman ran back and forth saying, ''Kill him 3/8''

Holt, who said he took the unusual step of releasing the statements because he would not allow anybody to call him or Prince a liar, also noted that several other witnesses said they did not hear those urgings.

In any case, residents have sent a strong message of support to police.

While the City Council pushed through minority recruitment measures Wednesday, drivers turned on their lights in the daytime, emblazoned their cars with ''Back the Blue'' bumper stickers and offered thumbs-up kudos to passing squad cars. City Hall was the scene of repeated rallies and candlelight vigils.

Prince, meanwhile, ordered officers to patrol in pairs for safety. How long that policy will last will be determined Monday.

''I would expect that would be the aftermath of this - that we could find a way to work together and go forward,'' Prince said. ''We have to make sure he didn't die for nothing. I hope his death represents a turnaround in public support for law enforcement - not just in Dallas, but all over.''

The impact of the shooting, and whether it will stall the push for reforms, has yet to be measured, Redlinger said.

''Obviously, you have a tragic death of an officer and it's being used to stage political theater,'' he said.

''I think it's very clear that our police department has been demoralized by a pattern of neglect of law enforcement,'' said U.S. Rep. John Bryant, D- Texas, who participated in the probe last summer. ''We now have (a department) that is 600 men short of a full force when compared to other cities.''

Prince said Friday he is ready to work with city officials on 81 police reforms the council adopted earlier.

Mayor Annette Strauss, mollified somewhat by Prince's pledge to work with the council, nevertheless publicly scolded him for blaming elected officials.

''I think the statements he made were divisive,'' she said. ''The worst part - the part that put Dallas in an unfavorable light - is that (his statement implied) there is a climate of hatred in this city. There is not.''

Dallas has prided itself on non-confrontational race relations all the way back to the civil rights movement of the 1960s. But in a city increasingly black and Hispanic - 29 percent black and 12 percent Hispanic, according to the 1980 census - the police department is 15 percent black and 7 percent Hispanic.

The racial makeup of the 2,460-officer force has drawn repeated attacks from two black council members and prompted charges of insensitivity to minority concerns and officers too trigger-happy with black suspects.

That didn't dampen last week's outpouring of support for police, but ''stuff doesn't last very long in Dallas,'' said a police officer, who asked that his name not be used. Attitudes didn't really change, he said.

Chase was the 49th Dallas police officer killed in the line of duty, according to records dating back to 1892, Sgt. Carol Eberhardt said. FBI statistics show that nationally, 65 law enforcement officers were killed in the line of duty in 1987.