The best research topics are ones that are debated among scholars. Observing the debate can help you formulate your own opinions and contribute your own argument to the debate while allowing you to prove your own opinion. There are many points of scholarly debate and looking over articles and books on literary criticism can help you uncover them, however, below are a couple worth noting that would make good research topics.
One good point of scholarly debate is whether or not the women in this play can be viewed as submissive, powerless, male-dominated women who contradict feminist views, or if they are rather women in their own right, making their own decisions and accepting men on their own terms. Some modern feminist scholars view Beatrice's union with Benedick as submissive. They argue that Beatrice only saw her faults of being antagonistic and overly prideful because Hero, the submissive woman, pointed them out. However, other scholars argue that Beatrice recognized her faults on her own accord, not just because Hero pointed them out. Some modern feminist critics also call Hero's reunion with Claudio in the final scene of the play a terrible act of submission; however, others point out that Hero forgave him only because Claudio was sincerely repentant, which gives her personal power, rather than makes her submissive.
Another interesting point of scholarly debate is whether or not Dogberry is an important character or if he can instead be considered disposable. Some scholars argue that Dogberry is trivial and that the entire plot lacks purpose; however, other scholars show that Dogberry represents the entire heart of the play, mirroring the vanity and excessive pride found in all of the other characters. More importantly, by being the only character who unfolds Don John's treachory, despite his own intentions not to catch any criminals that night, Dogberry shows us just how inconsistent and inadequate the other characters in the story actually are, especially the rulers of Messina.
Examine the treatment of gender roles in the play and compare them to modern times.
Examine incidences where the mistrust of female sexuality arises, how characters deal with it and compare to modern times, not to mention other works of literature where such things come up (ie-Scarlet Letter).
Examine the treatment of cuckoldry in the play (a woman having an affair out of the marriage) and compare to modern times. Is cuckoldry considered just a shameful today?
Examine the prevalence of spying, eavesdropping and gossip in the play how it furthers the plot and whether or not Shskespeare might be making a moral statement about the dangers of such things.