Trains are mostly liked people for their high-speed/ attractive-livery/ good-maintenance/ superb timings etc. Liking shall vary with region.
If we consider trains departing from Chennai only, (and leave rajdhani/shatabdi trains also) then most popular trains should be
at national level- Historically -GT express (till 1976)
at national level- TN express since 1976, for short distance- Brindavan is most popular, Vaigai Express also popular...
more... since introduction in 1980s
For TN state- Vaigai-Brindavan-Kovai-Pallavan may be top choices