Physical or mechanical sense: a place where something gets caught, a snag, or a hold that prevents smooth movement.
This drawer has a catch and won't open easily.
釣り糸に引っ掛かりを感じた。
I felt a tug on the fishing line.
Abstract sense: a personal connection, relationship, or tie, often with a nuance of something that links people or things.
彼とは何の引っ掛かりもない。
I have no connection with him at all.
この事件にはいくつかの引っ掛かりがある。
There are several leads in this case.
unease; discomfort; something off
Emotional sense: a feeling that something is not quite right, a mental snag or discomfort.
彼の説明にはどうも引っ掛かりを感じる。
I feel something off about his explanation.
その提案に引っ掛かりを覚えた。
I felt uneasy about that proposal.
Rare variant using 懸 instead of 掛.
Rare variant combining 懸 and り.