I have created a webservice in .net 2.0, C#. I need to log some information to a file whenever different methods are called by the web service clients.
The problem
You don't say how your web service is hosted, so I'll assume it's in IIS. I don't think the file should be accessed by multiple processes unless your service runs in multiple application pools. Nevertheless, I guess you could get this error when multiple threads in one process are trying to write.
I think I'd go for the solution you suggest yourself, Pradeep, build a single object that does all the writing to the log file. Inside that object I'd have a Queue into which all data to be logged gets written. I'd have a separate thread reading from this queue and writing to the log file. In a thread-pooled hosting environment like IIS, it doesn't seem too nice to create another thread, but it's only one... Bear in mind that the in-memory queue will not survive IIS resets; you might lose some entries that are "in-flight" when the IIS process goes down.
Other alternatives certainly include using a separate process (such as a Service) to write to the file, but that has extra deployment overhead and IPC costs. If that doesn't work for you, go with the singleton.