Most of the VF pages in my company org use remote actions, so in debug log the operation is coming as VFRemoting. I need to open many log files to find out the Apex class that has the remote action method I am interested in. For ex, If I need to view the Sytem.debug value recorded using a remote action in controller A, I need to open a few log files and apply filter for Controller A. Is there a way available to search for a word in all the log files shown to me in the log section?
It isn't currently possible (Summer '17) to use the native tooling to perform a search over the contents of all the log files.
You can do SOSL searches over the body of Apex classes and Triggers, but that isn't going to find you content in a Debug Log.
Searching over the Debug log content would require pulling the bytes of each log down and then searching them locally.
Update: I'd seen some posts recently about using the SFDX command line and GREP over the last N logs. It will still need to pull down the entire log(s).
See Streaming Debug Logs to your console