I have been watching high Elo stream for about 4 months now, I am myself a 1+ Year playing League Of Legends, and one thing i noticed, is that Lich Bane is not that famous, from low to high elos. Me myself love this piece really much, I can't see problems on it, and I don't know why people say that is not viable. A Bad-ass Karthus or maybe a Vlad, could with almost 500 AP, deals its skills dmg ignite, and can still miss some burst, why not only one auto attack, I know that are items like Death Fire grasp, but it has a longer Cd and might deal less dmg, and being MR more expensive than Armor, most of people have even less Mr, so a lich bane in my opinion can add you extra burst, but can also change your game completely.
Please post your opinions, because there is a reason people don't use it, and I need to know why.

Lich Bane is extremely expensive for what it does and is a great distraction from other, better items for most AP casters. Any AP champ who needs to get up close and personal (Akali, Kassadin) can use Lich Bane to great effect since they'll be in close combat anyway. Building it on a further ranged AP champion is generally a waste, mainly because you're never guaranteed that autoattack. Furthermore, Lich Bane is 3470 gold; Rabadon's Deathcap is 3600 gold. For nearly every AP caster in the game, stacking Rabadon's is far more effective than getting a Lich Bane because the massive AP bonus applies to most if not every skill the character has, not just to the autoattack.

LB is really only viable on hybrids when they stack AP (AP Ez, Akali, Jax spring to mind). For regular casters, Rabadon's is much better.

takes too long to get the AP needed to where lichbane would be good

Iv always considered lichbane on swain. With his dmg buff from torment i have always felt like it would work. But at the same time, it does take a while to farm up.

I use it on hybrid AP/Mana ryze when there is some reason to max E first. (E.G. the enemy has a pet [Malzahar] or innate resistance [Soraka, Kassadin, Jax])

AP + Mana + MR + Movespeed (Movespeed is underrated, as it is an offensive and defensive stat)

Had a very long drawn out match once where I had 3 death caps, a lich bane, sorc shoes and a hourglass on an Annie, it was pretty funny to watch people's health start to drop in an q + a auto attack, but then again, this was on bots, was 4v5, and had plenty of farming time, and nothing better to do, so it was just fun to watch xD

I get a lich bane on Sona after i have all my wards/suport items. The extra movement speed combined with sona's "E" and the runspeed clicky item and boots make you extremely fast. The attack passive is a beast on Sona as well.

In dom when i go full AP sona its my second buy after Death cap.

It synergizes extremely well with Lux's passive, which makes it one of her core items.