Update: Procured my missing volumes, so this should be more or less complete. I may be missing an apology or two, as I'm only human, so feel free to contribute if you think I missed one.
Volume 2
No apologies here
Volume 3
No aplogies here
Volume 4
Ranma apologies for kissing Akane (in the Neko-ken)
Volume 5
No apologies here
Volume 6
Akane feels incredibly bad about how she didn't listen to Ranma(dojo destroyer), and apologizes several times with tears in her eyes (Note: she was willing to practically do anything to make it up to him)
Volume 7
No apologies here
Volume 8
Akane apologizes to Ryoga, because he got hurt when Ranma kicked him (Ryoga was fighting Ukyo)
Ryoga feels guilty because he thought he betrayed Akane (Ranam dressed up as Ryoga's fiancee)
Volume 9
Tsubasa felt bad for how he treated Ukyo, and thought he was a terrible person
Ryoga feels bad about neglecting "Yoiko" ^_^
Volume 10
Akane apologizes to Ranma for not believing him about the hypnotic mushroom
Akane apologizes to Ranma about the super strength soba
Principal Kuno apologizes for trying to cut the students hair. ^_^
Volume 11
Akane apologizes several times for bashing Ryoga (waterproof soap date)
Ranma and Genma use the "Crouch of the Wild Tiger" on Akane. ^_^ (training for Hiryuu Shoten Ha)
Ryoga feels guilty for attacking a weakened Ranma
Volume 12
In the middle of the tornado, Ranma apologizes to Akane, for losing the chart
Ghost cat apologizes
Principal Kuno apologizes to Akane for not being able to teach her to swim
Volume 13
Ryoga feels bad about having the battling God Tattoo
Gambling King apologizes to Ranma
Volume 14
Akane apologizes to Ranma in the sandpit of the obstacle course race
Volume 15
Soun apologies to Ranma for getting him involved inthe Picolet Chardin thing
Akane apologizes several times to Nabiki for getting soy sauce on her jacket. Turns out the jacket was Akane's, and Nabiki doesn't feel apologetic at all for having taken numerous pieces of Akane's clothing.
Akane feels guilty about how she broke up with Ranma
Ranma apologies to Akane by the canal fence
Akane feels it's mostly her fault, and apologizes to Ranma by the fence.
Note: At this point I'm switching over to the Japanese numbering of volumes, which has Taro showing up for the first time.
Volume 18
Happosai begins to regret giving Pantyhose Taro his name ^_^
Volume 19
Ranma feels guilty about messing up the sauce
Ranma tries to apologize to Mr. Tendo, but apologizes to Akane, and then wonders why he's apologizing.
Ranma still feels guilty about the sauce, and apologizes to Ukyo for ruining it (Ukyo doesn't believe him, thinking Akane put him up to it)
Volume 20
Nobody apologizes in this volume
Volume 21
Akane feels guilty about the Sakura-Mochi she made
Volume 22
Akane apologizes to Ranma for "meddling" in his affairs (she was trying to get him to meet his mother as a guy)
Volume 23
Akane apologizes to Nodoka for messing up in the kitchen (three times), the last with tears in her eyes
Ryoga feels guilty about using the fishing rod
Volume 24
Akane apologizes to Ranma, because Ranma was locked in female form (he fought Herb because Herb tossed Akane) She feels it was her fault.
Volume 25
Ranma apologizes (for jumping off little Hinako and knocking her out, I presume)
Ukyo sort of apologizes to Hinako and wants to make up for it, but it was only a ruse to get her guard down
Shinnosuke apologizes to Akane for walking in on her in the bath
Volume 26
Akane apologizes because she thinks the food she made was bad
Akane apologizes again for the bad rice balls she made.
Shinnosuke apologizes because he thinks it's his fault the rice balls are bad (thinks he's sick)
Akane apologizes for slapping Ranma(Asks that Ranma hit her back)
Ranma apologizes for bothering Akane immediately afterward
Akane apologizes to Ranma for making him worry
Volume 27
Kuno apologies to Hinako ^_^
Soun apologizes to Hinako because it looked like Ranma was taking advantage of her in his house
Ranma apologizes to Shampoo for turning down her date request (He doesn't sound apologetic, though: "Sorry. Gotta go.")
Ranma spologizes to Mousse many times with tears in his eyes (Glasses of Invincibility ^_^)
Mousse feels guilty about using the glasses, begs Ranma to beat him up.
Ranma apologizes, because he doesn't want to do it ("Sorry. Not my style.")
Ranma apologizes to Mousse due to the glasses again
Shampoo apologizes, because she won't do anything about Mousse due to Akane's request to help. ("Sorry. No way.")
Mousse apologizes to Ranma(he put glasses on backwards, but did he really mean it?)
Kuno apologizes for not flattering the swimsuit
Ryoga feels guilty for leaving Akane behind in the Tunnel of Lost Love
Ryoga apologizes to Ukyo for not keeping with the plan
Akane apologizes to Ukyo, for not suspecting Ukyo was in love with Ryoga. ^_^
Volume 28
Nobody apologizes in this volume
Volume 29
Kuno apologizes to the pig-tailed girl for not being able to remove the phoenix from his head ^_^
Volume 30
Ranma apologizes to his father, because the hair growth potion didn't work
Genma apologizes to Ranma for stabbing Ranma with his hair while asleep ^_^
Akane apologizes to Ryoga for getting him mixed up in the cursed spatula business.
Volume 31
No apologies in this volume
Volume 32
Ranma feels guilty about trying to "seduce" Akane, to get rid of the suit (feels guilty that Akane is being nice and agrees to get rid of it)
Akane feels bad for allowing herself to fall for Ranma's trick
Ranma apologizes to Akane for trying to seduce her(playing with her feelings)
Volume 33
Densuke knows it was wrong to put sleeping potion in Ranma-chan's drink, in order to have his way with her (yet does it anyway ^_^)
Family tries to apologize to Kasumi, for making her angry (Doesn't quite work...)
Ranma apologizes to Ryoga because the mushrooms are ruined
Volume 34
Akane feels guilty for not giving the chocolates to Ranma
A couple of guys apologize to Hinako for ticking her off. ^_^
Ranma apologizes for landing in the changing room while Akane is trying on a new bra
Ranma apologizes to Akane (probably for making fun of Akane's chest, though he doesn't really know what he did wrong)
Akane feels guilty about using Ranma's mother to get back at him (said she shouldn't have done that)
Akane apologize to Ranma for having "such a boring body" that Ranma would find it uninteresting
Shampoo feels a little guilty for washing Mousse's robes with the glasses still inside (she ruined the glasses).
Volume 35
Akane apologizes for being fat (of course, she's yelling it at the time while throwing a heavy object...)
Ranma-chan mirror clone aplogizes for causing trouble (she doesn't mean it, though)
Ranma feels guilty about ruining Nabiki's ticket(Nabiki takes advantage of that to make Ranma do many things)
Volume 36
Ryoga apologizes to Akari, for taking so long to get to their date.
Genma apologizes to the unconscious Nodoka (although he used knock-out gas on her and was interested in the Saotome treasure...)
Volume 37
No apologies in this volume
Volume 38
Ranma apologizes to Akane, for being so awkward with words (she's unconscious... or is she?)
Number of times person apologized or felt guilty (based on above):
Akane=21
Ranma=19
Ryoga=7
Everyone Else <=2
Infer from this what you will, and feel free to e-mail me with apologies (or feelings of guilt) that I may have missed. I'll most likely include them, and let readers decide how legit the apology and/or feelings of guilt were. Try to keep it to feelings of guilt over something that the character thought was their own fault, though(usually own actions, though a character could infer that their actions indirectly caused something to happen). The volumes I'm missing, I have on order, so I'll add stuff from them when I receive them.
My thoughts? Signs of guilt, and heartfelt apologies, are a sign of maturity, character growth, and most importantly, a conscience. For a manga where people claim that everyone remains static and/or immature, this is my way of saying, "Hey, some characters can rise above their shallow characterizations and be good people." Incidentally, this is one of the big reasons why I like Akane and Ranma, because of their repeated times when they evidence a conscience. Though, I do like everyone in the manga, because at heart it is a comedy, and perfect people in a comedy would be considerably less-than funny.
Again, my own thoughts, however. :)