This is what I've observed so far:
One of my comics which hasn't been updated in weeks appeared in Trending a couple of times when it got stumbled on by readers and received a handful of likes/subs/comments in a short amount of time.
As for Popular, it takes a little more time than trending for comics to show up on there (maybe around 5 to 12 hours later) after a newly added episode accumulated loads of views, likes and comments (subs not really necessary). This typically happens to comics that have 1000+ subscribers (and viewers combined) who swarm in to read the new episode or to comics that get featured on the daily snack.
BUT, they don't stay on Popular for too long, so that other comics that just updated have a chance. They stay up there for around 10 hours or so (I can't give an exact time because I don't look at Tapastic 24 hours) and then they're taken completely off the Popular list until the next new episode and wave of views/likes/subs -- even if the comic was in the middle of consistently getting plenty of attention through Popular. There are some comics that stayed up on Popular for days, like Mecha Necha and Scandanavia & the World, because they had episodes added continuously without skipping a day.