My knowledge isn't first-hand, but each school's reputation has some merit; most of these perceptions reflect stereotypes of the areas of town/neighborhoods that feed into the respective schools and almost always involve race (tacitly or otherwise) and the achievement gap --- which has and will continue to be an issue in Madison. I'll deal mostly with the stereotypes.
WEST Pros: West would likely be considered Madison's flagship school. Most of the UW faculty's children attend West, it usually has the most National Merit Scholars, future-Ivy Leaguers and overall "intellectuals." West kids seem to be the most well-rounded, interest-wise (it's not uncommon for a West kid to dabble in sports, music, art, drama, politics, academic clubs, etc. and not be thought any less of for it--- they're tough to pigeonhole...which is good). As a previous poster stated, West has a strong liberal arts program and a fair share of students end up going to elite liberal arts colleges. They're typically good in the "preppy" sports: soccer, tennis, swimming, cross country, club rowing, etc.
Cons: There's a large income and achievement gap. West draws from some of Madison's wealthier neighborhoods: Shorewood Hills, Vilas, Nakoma, Regent, Dudgeon-Monroe, Seminole Ridge/Highlands (Fitchburg) and also some of its "worst" (Burr Oaks, Bram's Addition, pockets in the Leopold area, etc.). These wealth/achievement disparities exist in all Madison schools, but West comes off as the most nose-in-the-air, in spite of its overall inclusive vibe. One could say West has a drug problem...it's rare to find a West kid who doesn't smoke weed. The school facilities are old and crowded from what I've seen.
Memorial Pros: Would likely be considered a close second behind West reputation-wise and many of West's positives apply to Memorial. Like West, a lot of highly-educated professionals' kids attend Memorial and they also have a fair share of National Merit Scholars and future-Ivy Leaguers. While Memorial kids on average wouldn't be considered as well-rounded as West kids...they have a little more /middle-class/grounded/blue collar (read: not elitist) mentality. The math/science program is to Memorial is what the Lib Arts program is to West. Memorial has elite swimming(!) and soccer programs --- and also basketball/hockey/baseball, depending on the year.
Cons: Copy and paste West's large income and achievement gap stuff, but substitute in Blackhawk, many areas around Old Sauk, and Hawk's Landing (wealthy) --- and Allied Drive and certain areas around Raymond Road (not-so-wealthy). Memorial seems to celebrate its racial diversity more, social intregration-wise ---although I've heard reports of parents bussing their children to far-away schools to avoid "problem" schools like Toki at the middle school level.
I don't know the reputations La Follette and East as well, so I'm eschewing pros and cons, however:
La Follette: Kind of like Memorial's poorer, widely-middle class, blue collar sibling. La Follette has character. Depending on the year, good CC, track and basketball programs. It doesn't really stand out any particular area but I'll gladly stand corrected.
Madison East: East probably has the worst reputation and the largest percentage of socioeconomically disadvantaged kids --- some neighborhoods around Northport are considered some of the worst areas in the city. East is like West in some aspects (the well-off liberal vs. poor dichotomy) although East's brand of liberalism is a little bit more granola-y, middle class, progressive and less elitist. Like West, East kids seem to be fairly well-rounded. East has elite tennis, track and basketball teams, depending on the year. From a few brief visits, East's facilities looked pretty rundown/outdated --- as the building itself is pretty old.
Edgewood: Pretty much how you'd envision a Catholic private school. Reasons (or combinations thereof) parents send their kids to Edgewood (A) they're Catholic (B) they can, financially ...as a show of status, like joining a country club (C) for a "better" education (D) to unabashedly avoid poor kids/diversity/"distractions"
You pay for what you get, and it's supposedly good college prep. They put a lot of emphasis on writing, literature, constructing essays, etc. Their facilities, food, teachers. are all top-notch. They take coach buses to sporting events. A lot of well-educated, well-rounded kids manage to come out of Edgewood. But there's the Catholic stuff. And a jarring lack of diversity. And unfettered elitism. And the 10k(?) a year in tuition. And a lot of seaworthy douche-canoes also come out of Edgewood.
Monona Grove = a less rural, slightly more diverse, more middle class/liberal Waunakee with more of a Madison feel/connection. In other words: a mid-sized school elite in some sports with wealthy neighborhoods near or on the lake. Anecdotal, but everyone I've met from MG has been pretty laid back, nice, hard-working and without pretense.
If was sending my future kids to Madison or Madison-area schools (thus excluding Sun Prairie, Verona, Waunakee, DeForest, Oregon, etc.) my rankings would be as follows: (1) West (2) Middleton (3) Memorial (4) East (5) Monona Grove (6) La Follette (7) Edgewood