For normal gameplay (for me) where I follow a family through generations, I always go after the Soulmate and various Family aspirations except for when I want a generation where things are more dysfunctional. So I'd have to say those are my favorites, as I do them on almost every sim.
I generally start a sim (as a teen) with a different, more skill/career oriented aspiration and try to finish that in their lifetime as well. My first generation in my current game, for example, the father completed the cooking one and the mother completed one of the creative ones (forgot which...it was 3 generations ago).
I switch aspirations pretty regularly rather than completing one before doing the next, though. For example, with soulmate you need to go on dates and do romantic aspirations, so when I have a sim working on that one I switch to it when they are with their SO and going on dates...but I switch it back to a different one when they are skilling/working/etc. Generally, manage to complete 3 or 4 aspirations total before a sim passes from old age doing it this way.
Other aspirations I do alot, once a sim household is established, are the two fortune ones. Those both are practically instant-completes with a household that has been around for a long time.
In my current game, the heir just aged up from teen and has already completed both fortune aspirations and the Fitness aspiration all while still a teen. She's working on the Location one now as a new YA, while ducking in and out of the romantic ones while doing romantic things. Her parents are adults, not elders yet, and have completed the same 3 aspirations as her, as well as Soulmates, one of the Family ones (and are near completion of the other 2 Family ones), and 1 or 2 other skill-oriented ones each.
You aren't wrong about the Successful Lineage though...it is the last of the Family one that I manage to complete, and most generations the grandparent sim dies before it is achieved. Even when they haven't died...the reward trait is basically useless. I can see it being more practical for someone who plays with long lifespan or aging off though, or if you turn that sim into a vampire (or ghost) and keep them around a lot longer.