The term "neurosis" is out of fashion in formal psychiatry. It was removed from the Diagnostic and Statistical Manual of Mental Disorders in 1980 and is now more of a colloquialism for quirky, anxiety-inspired behavior. Though, behaviors may go by different names, unusual behaviors result from high anxiety. They are not behaviors of a person who is psychotic (detached from reality.)
Obsessive Hand Washing
Obsessive hand washing or other obsessive behavior is marked by a ritual which feels like a requirement, but offers no real benefit. For example, the obsessive hand washer may feel the need to wash his hands repeatedly, perhaps in very hot water, to soothe anxiety or emotional distress, even if his hands are sanitary and don't need cleaning. The behavior, much like an addiction, only feeds the neurotic cycle of responding to discomfort with an ineffective, compulsive behavior.
Self Mutilation or Cutting
Cutting or other forms of neurotic self mutilation is often done by teenagers. The practice is generally not a suicide attempt, but a difficult-to-control neurotic response to extreme stress. The cutter injures herself, often to feel a direct correlation between cause and effect or control that seems absent from her life. Cutting is only a neurotic trait if the cutter is fully aware of the behavior, doesn't want to do it, but doesn't have alternative coping skills.
Phobias Such As Agoraphobia
There's nothing unhealthy about fear. In fact, it's a critical survival mechanism. A phobia is an irrational fear. It's a fear that doesn't have a basis in fact, or a fear that is excessive relative to actual risk. One example is agoraphobia. Sometimes described as a fear of public places or leaving your house, agoraphobia can be an overwhelming fear which invokes anxiety attacks any time the sufferer feels he cannot escape a situation.
Depression is not always thought of as neurotic, but depression and anxiety are closely related. Dysthymia is a persistent, low-level depression, often caused by persistent anxiety. Dysthymia is often accompanied by quirky and anti-social behavior. A common, neurotic trait of the dysthymic person is an unrealistic sense that he has a more acute understanding of the world.