Great to see all of the replies to the original question, and other questions raised in result!
personally, I remember a lot from my last lifetime. It was very brief (I believe I was born sometime around 1963 and died around 87, born in 1990 this life), and very traumatic. This is actually the story of "Elese", of whom I carried a lot over with me into this life, especially when I started my membership here (thus using that name instead of my current name):
I was a female from a very troubled family life (confirmed through memories I've had with a friend who was my mother and a "bad person" in this life also), my mother possibly being an outlaw (or had done something bad) and dying when I was young. I don't remember much between the time she died and the next big event, which came to me in a dream. From what I can guess by the symbolism in the dream, I was about 16 or so, and either almost raped or actually raped (not sure on, I lean towards actually) by a classmate I trusted, and I think that other of my male friends may have murdered him as a result (though this memory is all based on the dream I had where I recognized myself as her, so it's hard to distinguish what is memory and what is symbolism to validate that something terrible happened).I'm not sure if it was before this event or as a result of, but I became highly addicted to heroine and spent the remainder of my short life shooting up. I died from an overdose, and it's questionable whether it was intentional or not.
I think that we remember lives that are most important to our current lives the best because they are most relevant; for me, Elese has always been my reminder that I have those tendencies to fall into drugs/depression and how I have to face the hardships so I can become a stronger soul and move on from those lessons. Though I have never actually fallen on heroine in this life, I started my familiarity with substance abuse (alcohol and pills) from an early age in this life, mostly spanning from the time I was 12-15, and at some points getting so bad that I've been in the hospital as a result. I thankfully made it out of that trap mostly because I was able to recall Elese and what a fiasco that lifetime was.
Now I can hardly relate to the person I was as Elese or even just the four years ago in this life; though I realize I could always fall back to it, I now understand why I am drawn that direction and am able to avoid it with little difficulty and move on to other lessons.