Is there a list of must-have columns for LC-MS/MS used in biochemistry/clinical lab setting?
This answer will change fairly frequently because technologies are always improving.  These days, I think a bare minimum list might be something like:


a 2.1 mm &times; 50 mm C18 column for rapid reversed-phase methods.  
a 2.1 mm &times; 150 mm C18 column for slower, broader-coverage reversed phase methods.
a 2.1 mm &times; 50 mm HILIC column for rapid methods to detect polar metabolites.
a 2.1 mm &times; 150 mm HILIC column for slower, broader-coverage methods to detect polar metabolites.


Some further notes:


If your lab is going to use ion-pairing based methods, you will probably want separate columns for ion-pairing methods and "clean" methods without ion-pairing reagent.  Ion-pairing agents like tributylamine and/or perfluoroheptanoic acid are very difficult/impossible to remove from columns once they have been used for this purpose.
Fairly soon the best answer might be to use 1.0 mm diameter columns instead of the "standard" 2.1 mm diameter that is most common nowadays.  
The most widely used HILIC column seems to the "BEH Amide" column chemistry from  Waters.  Following close behind may be ZIC-HILIC from SeQuant and/or Agilent's HILIC-Z chemistry.  
For C18 chromatography, performance differences between different vendors are comparatively minor (although real).  A wide variety of chemistries are used.