This is my conclusion for the FRR based on Sarna.
http://www.sarna.net/wiki/KungsArm%C3%A9In addition to those listed as destroyed, some were destroyed, but later remade. I think this list is definitive as to FRR mech regiments.
1st Drakøns - Destroyed during Clan Invasion
2nd Drakøns - Destroyed during Clan Invasion
3rd Drakøns - "the tough and unforgiving battle ending with the destruction of the last Drakøn 'Mech two weeks later."
http://www.sarna.net/wiki/3rd_Drak%C3%B8ns#cite_note-WC-4 4th Drakøns - "With the Fourth's defensive positions compromised, the Drakøns were effectively obliterated in a holocaust far too similar to the destruction of Rasalhague."
http://www.sarna.net/wiki/4th_Drak%C3%B8ns 1st Freemen - Destroyed during Clan Invasion
3rd Freemen - Destroyed during Clan Invasion
1st Hussars - Destroyed during Clan Invasion
2nd Hussars - Destroyed during Clan Invasion
1st Kavalleri - Destroyed during Clan Invasion
3rd Kavalleri - Destroyed during Clan Invasion
Gunzburg Eagles - Surrendered and disbanded
That leaves four regiments at least partially intact
2nd Freemen - actually fought effectively
http://www.sarna.net/wiki/2nd_Freemen_(Clan_Ghost_Bear)
2nd Kavalleri - "Only five companies of the Kavelleri were able to escape from the planet and fall back to the world of Ueda."
http://www.sarna.net/wiki/2nd_Kavalleri4th Kavalleri - did not see action
3rd Hussars - did not see action