To argue about an inconsequential and trivial aspect of an issue
When you are accused of being forty-five minutes late for an appointment, you are splitting hairs to say that you were really only forty minutes late
When a man is about to urinate, and a hair or lint is resting on the opening to the urethra. This causes the sream of urine to split off in 2 seperate directions and, if not at a urinal, all over the bathroom. This will be to the much displeasure to the wife/girlfriend/other family member.
I had to take a huge piss this morning, but i had a splitting hair and hosed the entire toilet. The wife made me scrub the whole bathroom for 3 hours.