Sir first off, let me clear this that a Honda City is NOT in the same category as a Corolla. Corolla is at least 20% larger (although still has less cabin space, talking about the 02-08 model). Honda's competitor to corolla is Civic and thats much better then corolla. City's competitor in the toyota range is the Belta!! which was never introduced in local market. Belta and city have more or less same size, same features (abroad) and same engines too.
Having said that, i must tell u that I ve driven the 02-08 corolla extensively and i never found it to be any more stable then my city. infact brakes were worse and compromised stability if applied at high speed. the current generation of corolla, however, is better then the idsi (but NOT new city) in terms of stability.
as for axio and fielder, i ve driven the axio a few times in isb and yes i found it to be better then the local corolla but i never understood why its better. may be they used stiffer suspension or sumthing but yes, its better. cant say anything about fielder. and btw, an axio is technically a new corolla too, so u cant compare it to an idsi or even new city.