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The most precise realization of inertial reference frame in astronomy is thecatalogue of 212 defining extragalactic radiosources with coordinates obtainedduring VLBI observation runs in 1979-1995. IAU decided on the development ofthe second realization of the ICRF2 catalogue. The criteria of best sourcesselection (in terms of coordinates stability) must be defined as the first aim.The selected sources have to keep stable the coordinate axes of inertialastronomical frame.  Here we propose new criteria of source selection for the new ICRF catalogue.The first one we call as "cosmological" and the second one as "kinematical".The physical basis of these criteria is based on the assumption that apparentmotion of quasars (at angular scale of the order of hundred microarcseconds) isconnected with real motion inside quasars. Therefore apparent angular motioncorresponds to real physical motion of a "hot spot" inside a radio source. Itis shown that interval of redshift $0.8 \div 3.0$ is the most favorable interms that physical shift inside such sources corresponds to minimal apparentangular shift of a "hot spot". Among "cosmologically" selected sources wepropose to select motionless sources and sources with linear motion which arepredictable and stable over long time interval.  To select sources which satisfies such conditions we analyzed known redshiftsof sources and time series obtained by our code and by different centers ofanalysis of VLBI data. As a result of these analyses we select 137 sources as abasis for the ICRF2 catalogue.

Cosmological and kinematical criteria for the ICRF2 sources selection