Searching and filtering with KnockoutJS in C# MVC

KnockoutJS is a superb companion for client-side data-binding. Sometimes however you need to do something, and while there are numerous examples out there on the great JSFiddle etc, there is no explanation of how the code works, the docs don’t go deep enough, and one is left head-scratching to work out why things work as they do. I was recently trying to get a solid solution for implementing a combination “filter and search” on an observable array in Knockout. The data from the array is received by the browser from the server using ajax.