(Long post ahead. Tl;dr, projection.)

The mane six are memorable and lovable because each of their personalities is highly exaggerated, but still within believability. It's probably a bit of a stretch to actually "diagnose" (how can you diagnose a fictional character?) them with personality disorders, and we all know there's a lot more to their characters than this, but here goes:

Rarity and Rainbow Dash - Narcissistic Personality Disorder
This is probably the most obvious one. One of the first things that comes to mind when describing these two is "attention-seeking." The DSM-IV diagnostic guidelines for NPD:

A pervasive pattern of grandiosity (in fantasy or behavior), need for admiration, and lack of empathy, beginning by early adulthood and present in a variety of contexts, as indicated by five (or more) of the following:
1. Has a grandiose sense of self-importance (e.g., exaggerates achievements and talents, expects to be recognized as superior without commensurate achievements)
2. Is preoccupied with fantasies of unlimited success, power, brilliance, beauty, or ideal love
3. Believes that he or she is "special" and unique and can only be understood by, or should associate with, other special or high-status people (or institutions)
4. Requires excessive admiration
5. Has a sense of entitlement, i.e., unreasonable expectations of especially favorable treatment or automatic compliance with his or her expectations
6. Is interpersonally exploitative, i.e., takes advantage of others to achieve his or her own ends
7. Lacks empathy: is unwilling to recognize or identify with the feelings and needs of others
8. Is often envious of others or believes others are envious of him or her
9. Shows arrogant, haughty behavior or attitudes.

Rarity
Grandiosity is clear; if anything, that's the main reason we love her. She's so over-the-top in her refinement that it becomes ridiculous. Number six is shown a few times throughout the series, when she charms others into doing tasks for her. Her taking advantage of Spike's love for her is used as a running gag, and she was also shown to use her feminine charms (and think nothing of it) when she asked those two stallions to pull their carriage to the GGG.

Number two is also quite clearly shown throughout the series. It was made clear that her whole purpose for going to the GGG was to find ideal love. She's also become preoccupied with success and fame on numerous occasions, mainly "Green is not your Color" and "Sweet and Elite" (though this character trait was the basis for those two episodes, it should also be mentioned that she overcame that character trait by the end of the episode).

Numbers one, three, four, and five can be inferred from the show, but are never made into explicit character traits. Of those four, numbers four and fire seem the most likely; entitlement being shown especially in Rarity's avoidance of anything dirty.

Rainbow Dash
RD is also shown to be preoccupied with fantasies of success and power in her arrogant assumption that she'll become a Wonderbolt. (For her, it's not a hope or a dream, but a fact--only a matter of time.) In all, RD probably shows the traits associated with NPD even more clearly than Rarity. Number nine in particular (and number 7 to a slightly lesser degree) is an integral part of her personality.

Pinkie Pie - Histrionic Personality Disorder
DSM-IV:

A pervasive pattern of excessive emotionality and attention-seeking, beginning by early adulthood and present in a variety of contexts, as indicated by five (or more) of the following:
1. is uncomfortable in situations in which he or she is not the center of attention
2. interaction with others is often characterized by inappropriate sexually seductive or provocative behavior
3. displays rapidly shifting and shallow expression of emotions
4. consistently uses physical appearance to draw attention to self
5. has a style of speech that is excessively impressionistic and lacking in detail
6. shows self-dramatization, theatricality, and exaggerated expression of emotion
7. is suggestible, i.e., easily influenced by others or circumstances
8. considers relationships to be more intimate than they actually are

Number three and number six are the most obvious of these. In addition:
(4) Her outfit in "Dragonshy," etc.
(5) "And that's how Equestria was made!," etc.

Number eight is shown in "A Friend in Deed" by her assumption that everyone is her friend. Until Cranky Doodle makes is absolutely clear that he is not her friend, she assumes him to be. Given that the show is about defining friendship as something more than ordinary, this seems to fit with considering a relationship "to be more intimate than [it] actually [is]."

Fluttershy - Avoidant Personality Disorder
DSM-IV:

A widespread pattern of inhibition around people, feeling inadequate and being very sensitive to being evaluated negatively, since early adulthood and occurring in a range of situations. In addition, four of seven specific criteria should be met, which are:
1. Avoids occupational activities that involve significant interpersonal contact, because of fears of criticism, disapproval, or rejection
2. Is unwilling to get involved with people unless certain of being liked
3. Shows restraint within intimate relationships because of the fear of being shamed or ridiculed
4. Is preoccupied with being criticized or rejected in social situations
5. Is inhibited in new interpersonal situations because of feelings of inadequacy
6. Views self as socially inept, personally unappealing, or inferior to others
7. Is unusually reluctant to take personal risk or to engage in any new activities because they may prove embarrassing

Do I really need to expound upon this one? The main difference between this personality disorder and general shyness is the presence of the general personality disorder traits (see my post in the Personality Disorder thread). To be honest, the only way we'll know for sure how well those diagnostic guidelines fit with any of the characters is if we learn more about their childhood.

Twilight - Schizoid Personality Disorder
Oh, boy. This is a big one. SPD is a big, complicated thing. SPD is mostly defined by how the person perceives the world around him/her, and therefore can't be determined to any degree of certainty from the show. I mainly picked it because it "seems to fit." Maybe Cave Johnson can explain why I decided to give Twilight SPD:

"I'm no psychiatrist, but coming from a bunch of eggheads who wouldn't know the thrill of danger if it walked up an snapped their little, pink bras, that sounds like projection. They didn't fly into space, storm a beach, or bring back the gold! No, sir! We did--it's you and me against the world, son. I like your grit."

Applejack - Normal
As far as I can tell, Applejack is the only one whose personality doesn't fall in line with a specific personality disorder. Could be that my perception is colored by the area I live in (farm, ranch country), but the ICD acknowledges that personality disorder diagnostic guidelines are culture-specific.

Maybe that's why Applejack tends to be the least-liked.