As for middle schools there is: Sherman Oaks Ivy Bound Academy of Math, Science & Technology Charter Middle School. Though not a Charter school Robert A. Millikan Middle School is highly sought after for being an excellent school.
As for high schools both areas are known for two private schools, Campbell Hall and Harvard-Westlake. If you are looking for public schools I agree with Grace that Cleveland is well known as a magnet school though it is a distance from Sherman Oaks and Studio City.
I hope this helps,
Realtor, Berkshire Hathaway HomeServices
Sherman Oaks, CA