Why did citizens of Thebes not investigate Laius' murder at the time it occured?
When Sophocles' Oedipus the King opens, we find the people of Thebes suffering under the ravages of a terrible plague. Oedipus sends his brother-in-law Creon to Delphi to consult the oracle about a remedy for the situation. When Creon returns, he reports that the oracle has told him that the killer of Laius, the previous king, is still present among the Thebans. To rid themselves of the plague, Creon tells Oedipus that they must drive Laius' killer from their city.
Upon hearing this, Oedipus wonders why the Thebans did not try to investigate Laius' death at the time it happened.
When the ruling king had fallen in this way,
what bad trouble blocked your path, preventing you
from looking into it? (Ian Johnston translation)
Creon, in turn, responds that at the time the Thebans were also faced with another problem, the Sphinx, whose death, ironically, Oedipus subsequently brought about by solving the monster's riddle.
It was the Sphinx—
she sang her enigmatic song and thus forced us
to put aside something we found obscure
to look into the urgent problem we now faced.
Simply put, the Thebans have other things on their minds, most notably the Sphinx. This terrifying monster has put a curse on the city and she's warned the citizens of Thebes from inquiring into the circumstances of Laius's death. Not surprisingly, the Thebans are extremely reluctant to defy this mysterious creature. And in any case, only one eyewitness of Laius's murder remains—an old man crazed with fear now hiding out in the mountains. As he's gone completely mad, it's unlikely that he'll be able to provide much in the way of reliable information, certainly nothing that could help solve the mystery.
Strictly speaking, it would be relatively easy to discover the murderer's identity, were there a will to do so. But it's not really in anyone's interests to do this for the reasons just mentioned. Ironically, the one man who believes it is in his best interests to uncover the killer's identity is, of course, the killer himself.