Before I get started I'd like to clarify that these are all within the context of support Lulu. I have no issue with Lulu in other lanes, but it's not what I'm focusing on here.
I feel like the main reason behind Lulu's underuse and her occasional perceived weakness when she is played can be largely attributed to misunderstandings and/or disagreements about what her playstyle, strengths, and weaknesses are.
I feel a big contributor to this factor is that she can in fact be played a handful of ways in the support role. For example, I find many people tend to play her like Janna during the laning phase, focusing on shielding the carry and harassing with Glitterlance, while saving whimsy for when a fight is committed to (this part I wholeheartedly agree with), while others (such as myself), tend to focus more on starting harassment with Help, Pix! in order to land an easier and less predictable Glitterlance on one or both enemies. I don't know that either is necessarily better than the other (it probably really depends on the lane you're facing), since the main reason I go for the latter is honestly more for the sake of keeping myself entertained than because it necessarily contributes more to winning the game.
These differences may seem miniscule, but they heavily affect player's perceptions of Lulu's strengths and weaknesses. My approach tends to have ridiculous trouble against Sona -- I get in close enough to land a Help, Pix!, only to take more damage from Sona's abilities before I'm able to get out, forcing me to play passively while Sona easily heals off all the harass I do manage to get off. I'll get more into why I perceive Lulu to be weak against Sona later.
(TL;DR): Moving on to the actual point of this post, what are your personal perceptions of Lulu in terms of her strengths, weaknesses, matchups, etc? Whether these are perspectives gained from playing her, or from simple theorycrafting, I'd like to hear them (though I'd prefer you tell me if it's solely from theorycrafting), that we might nail down a more concrete idea of Lulu that could be useful.
I'll start it off:
As I've said, I take the more harassment-centered approach. My Lulu playstyle during laning revolves around foiling kill attempts (any one of her skills can do this, glitterlance only in some situations), keeping control of the bushes (with pink wards when necessary), and punishing enemies with E>Q whenever possible. As such, I typically max E first, usually leaving Q at rank 1, since I tend to use it more for the CC than the damage. After 6, if we haven't been losing lane, I'll typically play more aggressively and encourage my carry to go for a kill or two with the extra CC and safety that comes with Wild Growth.
This is pretty standard and straightforward. On a teamfight we're not losing, I'll tend to use Wild Growth to CC more enemies than to necessarily protect someone; but in a teamfight that's not going as well I'll obviously use it on someone who needs it to survive. I tend to use Help, Pix! for vision rather than as a shield whenever I'm against a champion that stealths in the middle of a teamfight ( Kha'Zix, Rengar, Vayne, Talon, Shaco, Akali) or in a teamfight where brushes are likely to be a major factor.
Lulu's strengths are pretty basic -- decent harass, very strong saving potential, and some killing potential. I don't think anyone will disagree with me here; the only thing people will likely disagree on is which strengths you should play to, and how they affect her matchups.
No sustain, squishiness, very vulnerable levels 1-4 or so. The lack of sustain is obvious. She can mitigate it somewhat with shield, but it's not foolproof. Fortunately, this means she can use her mana on other things, but it's a downside nonetheless. Her squishiness is pretty well mitigated by her CC, mobility, shield, health on ult, and my preferred build, but these require levels and/or ability to cast, so getting hit by a CC early can really screw you over. If you get killed before level 3 or 4, you won't even have whimsy to get back to lane quickly if you went the standard Q>E start to try to secure bush domination, and Lulu's 300 base ms will really hurt your laning.
In my experience, Lulu is strongest against relatively unreliable kill laners. By unreliable, I don't mean they're unreliable in that they'll rarely get you kills; but rather in that they won't be able to execute their combo all that easily (usually because it relies on a skillshot), so it'll happen relatively infrequently, meaning Wild Growth will typically be up when it does happen. Some prime examples of these are Leona, Blitzcrank, and Alistar. Obviously, Lulu's early weakness comes into play here -- you have to play extremely carefully at early levels here, as you would with any champion, but any missed skillshot or blown combo (or a successfully executed combo that was denied a kill thanks to Whimsy or Wild Growth) means you're free not only to farm, but also to harass the living hell out of them, since none of the aforementioned champions can respond well to harass other than by going for a kill (Alistar may seem to be an exception to this with his heal, but if you force Alistar to put ranks into and spam his heal, it means he's pushing his lane and draining his mana, making it even less likely for him to land his combo, and he'll be missing ranks in his combo, meaning even when he does land it, it'll require a lot more follow-up from his AD carry to result in a kill because of the lower damage). All three of these examples also have the entirely desirable quality of either going into or dragging someone into melee range to CC them. This is just great for Wild Growth, making it sometimes turn an enemy Rocket Grab or the like into a kill for your team.
As I said, Sona. I'll tentatively add on Taric and Pantheon as well. Sona I already explained, but as promised, I have more to say -- after completely shutting down your levels 1-5 with her counterharass, Sona becomes one of the worst terrors in the game for Lulu at level 6. Her ult is one of the few CC's in bot lane that's almost guaranteed to hit you along with your carry when used, meaning you have little chance to save either you or your laning partner when it goes down. This tends to completely counteract both your saving and your killing potential in lane, and it hurts. Bad. Lastly, unlike a lot of supports, Sona scales into late game relatively well (perhaps not as well as Lulu, but pretty damn well either way), so the fallback approach of turtling into Lulu's good late game is less effective against her than it is with, say, Soraka.
As for Taric, it's a similar story to Sona. He can really hurt your early aggression with Dazzle, and turn you into a fabulous kill, while simply healing off most of the harass you do. Taric also scales pretty well into late game, though not as well as Sona, so turtling against him may be the best option, and isn't an altogether bad one.
Pantheon is a very nonstandard pick for bot support, and I've only seen (and done) it a few times, but it can really mess with Lulu due to her early levels weakness. The main thing that separates Pantheon from Alistar, from what I can tell, is the damage his stun combo does at level 2. Pantheon with AD runes hardly needs any help from the AD carry to chunk your health bar, forcing you to recall, get zoned to hell, or worse yet, die. With a lot of early aggression, Pantheon and his lane partner can get several levels on Lulu and hers before they have the ability to respond to it. It doesn't take long for this effect to wear off, and careful play can avoid it, but I feel he's the champ that best takes advantage of Lulu's weak levels 1-4, so I'm including him here. Support Panth is a a blast, too. If you haven't tried it, it has my recommendation.
Obviously I'm not going to ask that you go into as much detail as I did; I just want to hear people's thoughts on the matter, and I obviously didn't want to put all of this into a comment.