Let’s first assume that there is no other reason that triggers the subjunctive (such as a conditional, politeness, etc., see below). In that case, one needs to distinguish between können and the other modal verbs:

• For können, there is hardly any change of meaning whatsoever. In both, the indicative and the subjunctive II mood, this verbs expresses possibility. I would wager that the vast majority of subjunctive-II uses of können in the German language can be replaced by the indicative without any loss of meaning whatsoever (and can confirm this from my copy-editing experiences, where this replacement was a method to squeeze some space).

  Note, however, that when expressing very low likelihood, it is more idiomatic to use the subjunctive mood, e.g.:

  »Wie hat er das bloß bezahlt?« — »Er könnte im Lotto gewonnen haben.«

  The subjunctive mood is also sometimes used to express a lower likelihood than the indicative mood, but this only works when there is a comparison:

  Sie kann in ihrer Wohnung sein; sie könnte auch im Büro sein.

  Finally, in some contexts, using the subjunctive mood for können avoids it being confused as meaning to be able to or to have the opportunity to, e.g.:

  »Wie überquerst du den Fluss?« – »Ich kann schwimmen.«

  »Wie überquerst du den Fluss?« – »Ich könnte schwimmen.«

  In both cases, this can be understood as “I might swim”, but in the first case, it could also be read as “I am able to swim”.

• With dürfen, sollen, and müssen, using the indicative mood to express any sort of possibility has become rare and odd, except for some fixed or semi-fixed expressions. As a native speaker, I rarely ever use the indicative mood for these purposes. Take, for instance, your second example sentence:

  Der Täter darf nach etwas gesucht haben, was für ihn sehr wichtig ist.

  I am a native speaker, and while I acknowledge that dürfen can be read as to possibly be or similar here, it took me quite some time to get there. Instead I read it as to be allowed to.

Note, however, that there are often additional reasons that enforce the subjunctive mood. In some of these cases, the subjunctive modal verbs can indicate either possibility or their “regular” (non-possibility) indicative meaning:

• A conditional clause:

  Wo mag sie nur sein? Wenn nicht Sonntag wäre, könnte sie im Büro sein.

  Ihr Drucker ist ausgefallen. Wenn nicht Sonntag wäre, könnte sie ins Büro gehen und dort drucken.

  Be aware of implicit conditional clauses:

  Könnte er doch schwimmen! [Dann ertränke er nicht.]
  If only he were able to swim! [Then he would not drown.]

• Politeness:

  Könnten Sie mir bitte das Salz reichen?

• Implying that somebody did not do something despite having the ability, opportunity, allowance:

  Er könnte mit dem Bus fahren, aber läuft lieber.

  Sie müsste den Antrag heute einreichen, aber sie hat ihn nicht rechtzeitig fertiggestellt.

Let’s first assume that there is no other reason that triggers the subjunctive (such as a conditional, politeness, etc., see below). In that case, one needs to distinguish between können and the other modal verbs:

• For können, there is hardly any change of meaning whatsoever. In both, the indicative and the subjunctive II mood, this verbs expresses possibility. I would wager that the vast majority of subjunctive II uses of können in the German language can be replaced by the indicative without any loss of meaning whatsoever (and can confirm this from my copy-editing experiences, where this replacement was a method to squeeze some space).

  Note, however, that when expressing very low likelihood, it is more idiomatic to use the subjunctive mood, e.g.:

  »Wie hat er das bloß bezahlt?« — »Er könnte im Lotto gewonnen haben.«

  The subjunctive mood is also sometimes used to express a lower likelihood than the indicative mood, but this only works when there is a comparison:

  Sie kann in ihrer Wohnung sein; sie könnte auch im Büro sein.

  Finally, in some contexts, using the subjunctive mood for können avoids it being confused as meaning to be able to or to have the opportunity to, e.g.:

  »Wie überquerst du den Fluss?« – »Ich kann schwimmen.«

  »Wie überquerst du den Fluss?« – »Ich könnte schwimmen.«

  In both cases, this can be understood as “I might swim”, but in the first case, it could also be read as “I am able to swim”.

• With dürfen, sollen, and müssen, using the indicative mood to express any sort of possibility has become rare and odd, except for some fixed or semi-fixed expressions. As a native speaker, I rarely ever use the indicative mood for these purposes. Take, for instance, your second example sentence:

  Der Täter darf nach etwas gesucht haben, was für ihn sehr wichtig ist.

  I am a native speaker, and while I acknowledge that dürfen can be read as to possibly be or similar here, it took me quite some time to get there and feels very archaic. Instead I read it as to be allowed to.

Note, however, that there are often additional reasons that enforce the subjunctive mood. In some of these cases, the subjunctive modal verbs can indicate either possibility or their “regular” (non-possibility) indicative meaning:

• A conditional clause:

  Wo mag sie nur sein? Wenn nicht Sonntag wäre, könnte sie im Büro sein.

  Ihr Drucker ist ausgefallen. Wenn nicht Sonntag wäre, könnte sie ins Büro gehen und dort drucken.

  Be aware of implicit conditional clauses:

  Könnte er doch schwimmen! [Dann ertränke er nicht.]
  If only he were able to swim! [Then he would not drown.]

• Politeness:

  Könnten Sie mir bitte das Salz reichen?

• Implying that somebody did not do something despite having the ability, opportunity, allowance:

  Er könnte mit dem Bus fahren, aber läuft lieber.

  Sie müsste den Antrag heute einreichen, aber sie hat ihn nicht rechtzeitig fertiggestellt.