Miroku and Inuyasha get along very well together, although their relationship got off to a rather rocky start. Inuyasha didn't trusth Miroku at all and the two of them had agreed to get the Shikon no Kakera on a first come, first serve basis. Inuyasha was fairly rude to Miroku, with good excuse, considering that Miroku had tried to kidnap Kagome and steal the Shikon no Kakera that they had already collected. Miroku's perverse nature was also something that Inuyasha was rather weary of too. Once Miroku started travelling with the group, their relationship started getting better and Inuyasha learned to trust Miroku, even though the monk had some... less than favourable habits.
The relationship that Inuyasha and Miroku share is like that of brothers. Inuyasha's rash temper and habit of rushing into things without thinking of the consequences or the outcome makes him a younger brother figure to Miroku, despite the fact that Inuyasha is much older. Miroku is wiser and calmer, and is a good balance to Inuyasha's firey behaviour. Miroku is there to give Inuyasha advice and push him along in the right direction, though that doesn't stop him from playing around with Inuyasha with sarcastic remarks and rather delinquent suggestions. The relationship between these two guys has come along way since they first met! The two rely on each other very much and are always there for each other in times of need.
Although Kagome and Miroku get along, Kagome is a bit weary of Miroku's perverse nature. She's aware of what goes on in his mind and tends to subtly keep her distance from the letcherous monk by calling him "Miroku-sama." Of course, her formality is not only to keep a subtle distance between herself and Miroku but because Kagome really does respect Miroku for his hardships and his talented abilities. During the time that they have traveled together, Kagome has learned to accept Miroku's perverse nature (especially since he doesn't seem to hit on her much anymore) though that doesn't stop her from getting upset when he bungles up his relationship with Sango. Kagome is a girl with a big heart and she knows what it's like to be in love with someone who doesn't express his feelings very well... so she really feels for Sango in this sense and tends to show her frustration towards Miroku whenever he wanders off at the sight of a pretty woman.
Perversity aside, Kagome is deeply impressed by Miroku's abilities and knows that despite his delinquent nature, he's actually a very kind and caring individual. She respects this (although she doesn't approve of his delinquency). Like Inuyasha, Kagome is much like a younger sister to Miroku and he is constantly looking out for her, whether its for Kagome herself or for Inuyasha's sake. Kagome trusts Miroku and can rely on him to be there for her when things get rough and although Kagome would probably rather talk about her emotions to someone who can relate better, like Sango, Miroku's wisdom is appriciated.
With a habit of acting more like an adult that Inuyasha, Shippou's relationship with Miroku is a bit more mature sometimes but Shippou is still just a kid so he doesn't really understand Miroku's perverse intentions at all times. Though Shippou does seem to know what's going on in a situation more than Inuyasha does and, like Kagome and Sango, is disproving of Miroku's letcherous behaviour, if only because he knows Sango is being hurt by it. In any case, with Miroku's perverse nature aside, Shippou somewhat idolizes Miroku and Miroku is someone that Shippou probably wants to grow up to be -- Miroku is strong and always seems to know just what to do when the situation starts to get rough.
When Shippou isn't hanging around Kagome, he's hanging around Miroku. To Shippou, Miroku is an older brother who is there to help him with advice and look out for him and that's exactly what Miroku does. They have a fairly close, family-like relationship with each other, much like Shippou's relationship with Kagome.
See Love Life for more information.
Being a tanuki that many Inuyasha fans tend to forget about, Hachi has known Miroku longer than anyone else that Miroku is currently travelling with. To put it simply, Hachi is Miroku's partner in crime. Miroku often enlists Hachi to do his dirty work and when Miroku first met Kagome and Inuyasha, he used Hachi to create a diversion for Inuyasha. Miroku also tends to use Hachi to get girls and when worse comes to worse, Hachi is often enlisted to be a youkai of the hour when Miroku is looking for a place to stay (ie: Miroku often gets Hachi to impersonate a youkai to which he can exterminate in exchange for free board).
Although their relationship seems to be more or less...er, professional, for lack of better terms, Hachi seems to really idolize Miroku. At one point in the anime, Hachi went as far as impersonating Miroku to get women, food and a free place to stay. When caught, Hachi said that it was so fun being Miroku that he just couldn't stop. ;p
Appearing only a few times, Mushin is the monk that took care of Miroku when he was young. Mushin's habits are just as bad as Miroku's are and despite the teasing on both parts, Miroku really does care about Mushin and the feeling is mutual. Mushin seems to have been a friend of Miroku's family if not directly related, and knows quite a bit about the kazana. He happens to be the only one that can help Miroku out when his kazana might be torn open.
Unfortunately, because of Mushin's lack of appearances in both anime and manga, I can't really elaborate any further on how their relationship with each other is.