Is there possibility to set the default search domain name just like 'domain' statement in resolv.conf in *nix systems?

on the main routerboard, i don't think so, but the dhcp server could send this information to clients.

it could be useful to be able to set this information 'by hand' since its not common to configure core routers to receive their addresses via dhcp

I want to add my voice to the one of szuwar, even thought it's been a long time since his request.
Working on command line on local Mikrotik router, every time you need to specify a machine by name youd have to write, for example:
ping <machine-name>.<localdomain name>
even if it is on the same subnet of the router. On every *nix machine, from more than fifty years, there is an option to set the "default domain" do add to dns queries when not specified on command line, so you can write only:
ping <machine-name>
Even Windows has such an option, DNS connection suffix, that you can manually insert and, as szuwar noted, no one configure routers to receive their address via dhcp: it's mostly the router that distributes ip addresses via DHCP.
Could we make this an option request for future releases?

++1