I think of a flag carrier as one chosen to operate certain international routes back in the days when those were strictly controlled, typically with just one airline, designated reciprocally by each of the two countries, to fly the route.

Hawaii. IIRC they once had two "flag" carriers on the state level.

Wasn't there a semi-charter operation based at LUX? Or am I thinking of an earlier version of Icelandic?

Olympic (national carrier) which is now part of Aegean (private).

SAA apparently came close to going out of business.

Canada once had CP as well as AC. I always thought of both of them as flag carriers.

SABENA = such a bad experience never again.

SW ceased operating this year after 74 years in business.

So no more confusion with WN henceforward!

But Southwest also went de facto our of operation this year ....

You are right. Olympic should be on the list. I had some good flights on that airline, although as I recall their on-time performance wasn't so great. They had a nice lounge at ATH.

Sabena was a great airline! They had the best food of any Euro carrier. Even on short flights like BRU-LHR you got an excellent hot meal on Sabena. And the Qualiflyer program with old Swiss Air was pretty good.

Slovak airlines were a short-lived national carrier in the late 90s and early 00s.
Czech airlines, previously Czechoslovak airlines, previously Czechoslovak State Airlines are on life support. If we go back to interwar years, Ceskoslovenska Letecka Spolecnost was privately owned, but for a time handled all international traffic from Czechoslovakia. Ceased operation in 1939 and formally liquidated in 1945.

TAME Ecuador

Thanks. I had thought of SAETA (owned LAPSA though I'm not sure if that counts) but it was apparently held.

Agreed. That's what they were back in the day, whether or not they were government-owned. PA and TW were the two US "flag carriers" since they "carried the flag" to the rest of the world. Perhaps BN was also considered a flag carrier, with their South American routes, but I don't remember they're ever being referred to as such.

right, I should have thought of that one also; had a PDCS cert test flight on their 737 (which crashed into a mountain on approach to Cuenca less than two years after delivery)

BTS List
 

Don't know if anyone has mentioned these (or if they were truly Flag Carriers) but, using the Carrier Decode list from the BTS I find (Excel file attached):

CARRIER CARRIER_NAME START_DATE_SOURCE THRU_DATE_SOURCE AJ Air Micronesia Inc. 7/1/1981 12/31/1989 IR Iran National Airlines Corp 1/1/1995 12/31/2000 RB Syrian Arab Airlines 5/1/1996 12/31/2014 ST Belize Airways Limited 7/1/1981 12/31/1989 UMQ Air Manila Inc. 4/1/1994 5/31/1996 WM Windward Islands Airways 7/1/1981 9/30/1993 JU Yugoslav Airlines 1/1/1986 7/31/1996 SZQ Aerolineas El Salvador S A 1/1/2013 12/31/2020 ADQ Aeronaves Del Peru 11/1/1996 7/31/1997 WT Nigeria Airways Ltd. 11/1/1996 7/31/2010 VA Venezuelan International Airways 7/1/1981 12/31/1996 CP Canadian Airlines International Ltd. 1/1/1997 8/31/2001 GY Guyana Airways Corporation 1/1/1997 10/31/2001 AI Air-India 3/1/1997 10/31/2000 AM Aeromexico 7/1/1981 12/31/1989 AR Aerolineas Argentinas 5/1/1992 2/28/1997 AV Aerovias Nacl De Colombia 5/1/1997 11/30/1997 CA Air China 11/1/1997 9/30/2001 FI Icelandair 12/1/1991 1/31/1993 IB Iberia Air Lines Of Spain 1/1/1998 1/31/2001 JL Japan Air Lines Co. Ltd. 3/1/1996 12/31/1997 JM Air Jamaica Limited 1/1/1998 5/31/2001 KE Korean Air Lines Co. Ltd. 3/1/1996 12/31/1997 KL Klm Royal Dutch Airlines 10/1/1995 12/31/1997 LA Lan-Chile Airlines 1/1/1998 9/30/2005 LY El Al Israel Airlines Ltd. 1/1/2010 3/31/2017 MX Compania Mexicana De Aviaci 1/1/1998 12/31/2005 OA Olympic Airways 1/1/1987 12/31/1997 PH Polynesian Airlines Ltd. 4/1/1998 4/30/2004 PK Pakistan International Airlines 10/1/1987 12/31/1988 PR Philippine Airlines Inc. 10/1/1987 3/31/1998 PY Surinam Airways Limited 4/1/1998 12/31/2006 RO Tarom Romanian Air Transpor 7/1/1998 6/30/2017 SA South African Airways 1/1/1990 7/31/1994 SK Scandinavian Airlines Sys. 7/1/1998 2/29/2008 SU Aeroflot Russian Airlines 9/1/1998 11/30/1998 TA Taca International Airlines 11/1/1998 7/31/1999 TP Tap-Portuguese Airlines 12/1/1998 4/30/2001 UP Bahamasair Holding Limited 12/1/1998 10/31/2000 EOQ Aeroservicios Ecuatorianos 1/1/2000 9/30/2010 KU Kuwait Airways Corp. 1/1/2011 12/31/2011 EI Aer Lingus Plc 1/1/2007 12/31/2013 SV Saudi Arabian Airlines Corp 1/1/2001 12/31/2006 CIQ A/S Conair 1/1/2001 12/31/2006 MB Samoa Airlines Inc. 4/1/1996 12/31/2002 KB Haiti Air 7/1/1986 9/30/1987 MS Egyptair 1/1/2011 6/30/2012 NH All Nippon Airways Co. 4/1/1997 6/30/1998 MH Malaysian Airline System 4/1/2004 12/31/2004 GG Grenada Airways Ltd. 3/1/1994 9/30/2002 Q2 Zambia Airways Corp. 1/1/2005 12/31/2006 KA Hong Kong Dragonair 4/1/2006 12/31/2007 4H Belize Trans Air 2/1/2007 2/29/2012 UYQ Aerolineas Uruguayas S.A. 9/1/2018 3/31/2020

I'm pretty sure a number of these reincorporated and are probably wrong. I just filtered for and end date and selected ones with National Sounding Names. The entire list can be found here: BTS Transtats

"Flag Carrier" is an obsolete term these days. Any airline with traffic rights on an international route is technically a "flag carrier" on that route or routes. It had some meaning in the days when most airlines were government owned and few countries had more than one airline that operated international routes. With deregulation and the many open skies agreements, the term is largely meaningless now.

Actually it became quite meaningful under COVID. Flag carriers were subsidized by their respective governments to keep certain routes active during the pandemic. Such as BA, AF, etc. flying daily or at least a few times per week to JFK. Even if there were only a dozen passengers, AF served JFK throughout the pandemic. And other key routes. I know as I flew AF on the JFK route throughout the pandemic.

Dominicana de aviación from the Dominican Republic. Defunct in the mid 90's

Air Jamaica was partially owned by the government, was spun off, then went under.