The simplest binary insertion/deletion channel that I can think of is the following: with probability p, each bit independently results in two copies of itself. This is a special case of a subclass of channels that I have dubbed sticky channels, which are like sticky keyboards: each symbol can result in a random number of copies of that symbol.
Sticky channels have the nice property that contiguous blocks of (resp 1s) at the input correspond to contiguous blocks of 0s (resp 1s) at the output. This property makes sticky channels easier than more general insertion/deletion channels.
I've just had a paper on sticky channels accepted to Transactions on Information Theory; here's a link to a preprint. The main result is that for that simplest channel above, I can numerically obtain very tight bounds on the channel capacity. But of course I'd still like to know -- is there a simple formula that gives the capacity as a function of p? And is there are simple and efficient coding scheme that nearly reaches the capacity?