I've never personally seen an uncontested election at district or county level, and I certainly have the impression it is very rare at that level or higher. Once you get below that level though, my understanding is it is reasonably common.
It seems to vary a lot by council. Rural areas in particular seem to suffer from it more than urban areas. More spread out population?
Speaking of which, here's North Kesteven, post 2007 labelled ward map
here. The 2003 map may be a bit off as I had to reconstruct it based on parish boundaries, LGBCE Data and Individual ward maps in the Lead Dataset of deprivation data from the 2001 census. As a consequence,
here's the labelled ward map for that.
North Kesteven is another large chunk of rural Lincolnshire lying in the somewhat higher, though still relatively flat, west of county. As such it's very rural, mostly agricultural and home to several RAF bases. There are two settlements of note. North Hykeham was formerly a small village which has become an extension of the Lincoln Urban area and thus grown into a large town, though it retains a separate identity from Lincoln. Further south, Sleaford was the former seat of government of Kesteven County Council, and has long been the main focus of settlement in the area, with evidence of settlement going back to the iron age. It was primarily a market town for most of its history, but the arrival of the canals made it a minor industrial centre, with the economy primarily based in the export of agricultural produce.
Politically, the council has traditionally been too divided to produce a majority, with the main groups being Conservatives, Independent Councillors and the Lib Dems who are locally strong in North Hykeham. The Conservatives took full majority control in 2007 for the first time, and maintained it in 2011 where, against national trends, they increased their vote share and the Lib Dems have held on to their support. Meanwhile Labour have yet to recover since their vote collapsed in 2007.