Situational irony: The Stars
It is ironic that the very thing that made them consider themselves to be special is also the same thing that renders them to be just another Sneetch. When the other Sneetches get a star it initially diminishes the superiority the other Sneetches had over them.
Situational irony: Adding and Removing Stars
It is funny to the Sneetches go about adding stars to their body whilst the others remove them. It's ironic that neither groups can make their mind up about whether they should keep their stars to make them special.
Verbal Irony: Sylvester McMonkey McBean
Sylvester McMonkey McBean is a businessman who comes along to sell his tattoo service to the Sneetches. It is ironic that he refers to himself as the Fix-It-Up Chappie when in fact all he does is cause thinks to get worse.
Verbal irony: Star-On machine
Initially the Sneetches are very impressed by the Star-On machine as it seems like a one of a kind, life changing machine. Of course, this is all made redundant when McBean later introduces the Star-Off machine, to reflect the Sneetches changing situation.