I have experienced quite a number of ASI failures, all caused by blockages, and all in single pilot aircraft with only a single panel and pneumatic instruments.
I have had a number of bugs in the pitot tube causing a complete blockage - zero indicated airspeed, a little disconcerting, but quite straightforward - power plus attitude equals performance, the erroneous reading was obviously incorrect.
Bugs in the pitot tube causing a partial blockage - under reading by approximately 15 - 20 knots - very disconcerting as it was very difficult to recognize and confirm especially given that it was my first flight on an aircraft with this type of wing in over a month. Got airborne, kept nose down to build speed, went past office low and fast (office is alongside the runway), set a climb attitude and climbed away at a lower than indicated but higher than normal actual airspeed. Level flight and approach reinforced the diagnosis, with the confirmation being an off the low end of the ASI scale touchdown.
The third type of failure I experienced as a pax sitting in the RHS - a static blockage. Lift off speed indication was normal, and as altitude increased the airspeed indicator started to under read. A quick glance at the altimeter which indicated airfield elevation, and VSI which indicated 0 suggested a static blockage. On opening the alternate static valve the readings returned to normal.
As to procedure - it depends. The most critical time for accurate airspeed is approach and landing - cruise and descent without an ASI is quite straightforward (VFR). With experience in an aircraft the glances at the ASI only confirm what you already know from power and attitude. Personally I would avoid landing where there is a lot of turbulence and/or wind shear. Somewhere where there are engineers to fix the problem is always handy too.