In Act 1, Scene 4 of Hamlet, Marcellus famously declares that "Something is rotten in the state of Denmark." What other natural or "unnatural" events, imagery, or behaviors are used to describe the corruption of the Danish court or other aspect of the play?
The first unnatural event or image in the play is the presence of the ghost, Hamlet's deceased father. According to Barnardo, the ghost appeared when a star reached a certain position in the sky, "Had made his course t'illume that part of heaven," (I.i.35). This is also the first "supernatural" occurrence in the play (the ghost) corresponding with a natural image (the star in a certain position), which suggests a connection between the natural and supernatural worlds. In Act 1, Scene 1, Horatio remarks that the presence of the ghost foretells an impending disaster:
In what particular thought to work I know not,
But in the gross and scope of my opinion
This bodes some strange eruption. (I.i.66-68)
When the ghost appears in Act 1, Scene 4, Horatio and Marcellus warn Hamlet not to follow it. They fear the ghost will lead Hamlet to madness or to suicide, perhaps into the sea:
The very place puts toys of desperation,
Without more motive, into every brain
That looks so many fathoms to the sea
And hears it roar beneath. (I.iv.55.1-55.4)
In Act 1, Scene 5, Hamlet tells Horatio, "There are more things in heaven and earth, Horatio, / Than are dreamt of in our philosophy." (I.v.168-69). In other words, Hamlet is saying that the presence of the ghost has convinced him that there are more supernatural things and forces at work than they had thought. Hamlet invokes the supernatural forces again in Act 2, Scene 2, "'Sblood, there is something in this more than natural, if philosophy could find it out." (II.ii.350-51).
After watching the "play within a play," Claudius asks Gertrude how Hamlet is and she replies that he is as "Mad as the sea and wind when both contend / Which is the mightier." (IV.i.6-7).
In the final scene, following Ophelia's funeral, Hamlet directly addresses the political/cultural and spiritual (natural and/or supernatural) state of things:
Larded with many several sorts of reasons
Importing Denmark's health, and England's, too,
With ho! such bugs and goblins in my life, (V.ii.21-23)
Often, in Hamlet, when a character uses the word "nature," he/she refers to their way of life, the normal state of things. Anything "unnatural" could refer to anything abnormal (the king's murder, the ghost, Hamlet's "madness," etc.); the "unnatural" is not necessarily something abnormal in the natural landscape (weather, trees, etc.)