Once we get down to the level of neighborhoods, below the township scale, finding GIS data can be very tricky. Neighborhoods usually aren't officially recognized, at least not by any body large enough or interested enough in creating accurate GIS data descriptions beyond "the houses between East Smith Street and the river."
Online real estate firm Zillow has created this collection of 7000 neighborhood shapefiles and associated geodata. They're under a CC BY SA license so you can use them on anything as long as you credit Zillow and share them along.
This is now my first stop when searching for basic boundary shapefiles (state, county, town), USGS raster data, and TIGER data (2003). I'm not sure who runs this site but the simplicity and open licensing are greatly appreciated!
"Datamob highlights the connection between public data sources and the interfaces people are building for them. Our listings emphasize the connection between data posted by governments and public institutions and the interfaces people are building to explore that data."
There is a wide selection of innovative maps, datasets, and data visualization interfaces drawing on a variety of topics. You can spend quite some time going through the offerings here!
The Illinois Department of Public Health has an excellent dataset generator for every Illinois county and MSA across the state. In addition to disease and mortality statistics, you can integrate a wide variety of related demographic and socioeconomic data.
The US National Atlas website offers dynamic, interactive thematic maps, multiple map generators for creating your own custom maps based on available government data, as well as raw data available for download by GIS professionals.