For a perfect crime, murder in this case, you'd have to eliminate certain forensic elements (which I in no way profess to know in their entirety; but heres a few that stick out)
Most important, the will to kill without remorse. If your conscience wont let you do something like that, then you will subconsciously sabotage yourself. That, i believe would be the hardest part of pulling off the perfect murder. I dont know anyone (thankfully and hopefully) cold, calculating or downright evil enough to do something like that. I pride myself on being and associating with people, not monsters!
Motive: which you could eliminate, or at least shed doubt on by making it a completely random person with no ties to you at all
Opportunity: make it happen somewhere that it would be likely for ANYONE to have done it
Weapon: ive had many debates about the merrit of an "Icicle stabbing". no weapon, no prints, no receipts, etc
Obviously, there would be minute forensic evidence that couldn't be eliminated, like hairs, those pesky ever present carpet fibers, etc. But say it happened at a fair, or a carnival, or in a subway, somewhere where it wouldnt be unlikely for there to be countless forensic "red herrings" all over the victim.
The likelyhood of cold weather (a near necessity for an icicle stabbing) would probably mean that everyone would be in thick, insulating clothing, meaning you probably wont get hit with too thick a blood spray.
You couldnt do it right out in the open, but a darkened alley, or a rarely used bathroom could at least cut down on the likelyhood of witnesses.
And again back to that detatchment from humanity needed to kill another human. You have to be able to remain calm, to not run helter skelter (no pun intended) from the scene; to make sure the person is all the way dead or at least leave them in a very fatal condition so they can't identify you.
And the smarts enough to know that if you DID pull all of this off, you could NEVER tell anyone about it. You would have to be so inhuman that you would do all of this for the sole purpose of proving to yourself that you could do it, with no need to gloat or be boastful of your cunning.
I dunno. Thats just an "off the top of my dome" idea birthed when I saw this topic. Im not gonna do it, so feel free to pick apart at it or comment as you will.

Wont hurt my feelings. oh, btw, see ya in december!