Three clubs are currently sitting on 49 all-time trophies, FC Porto, Olympiakos, and Steaua Bucharest. It looks like Portuguese league leading Porto will make it to 50 this season and Olympiakos, currently 2nd in Greece, has a decent shot as well.
The next closest club in Africa is ASEC Mimosas of the Ivory Coast, who currently have a combined 48 domestic trophies.
The closest club in England is Liverpool, who have 45 domestic trophies to date, although Manchester United is right behind with 43.
In Austria, Rapid Vienna (48) has won the League 8 more times than Austria Vienna, but have 15 less cup wins than their rivals.
CSD Municipal and CSD Comunicaciones in Guatemala lead the way in the CONCACAF region with 37 and 35 domestic trophies each.
In South America, Colo-Colo of Chile (37), Peñarol of Uruguay (36), and River Plate of Argentina (32), lead the way.
I didn't dig very deep into the Asian countries, but the winningest club I could find was Al-Sadd in Qatar, who have 38 combined domestic trophies.
The Chicago Fire leads the way in MLS, with 1 MLS cup and 4 US Open cups. DC United and LA Galaxy are tied for second with 4 domestic trophies each.