MIA-CCS is operated non-stop by five carriers, American, United, AeroPostal, Avensa, and LanChilie with 9 dailies.
MIA-GRU is also heavily competed on, with American, United, VARIG, TransBrasil, and TAM each with dailies.
MIA-SJC has American, Iberia Airlines of Spain, LTU, MartinAir, and LACSA, though I don't think you can buy tickets for the LTU and MartinAir flights (those planes fly DUS-MIA-SJC-MIA-DUS and AMS-MIA-SJC-MIA-AMS; I think you can only tickets on the the MIA-AMS/DUS and AMS/DUS-SJC legs)
Also competitve from MIA...
MIA-BOG (Avianca, American, Aces, and LanChilie)
MIA-PTY (American, Copa, Iberia Airlines of Spain, LAB)
MIA-CUN (American, Iberia, Mexicana, AeroMexico)
MIA-PAP (American, Air France, ALM, H9)
And for those of you wondering, Iberia has a hub in MIA serving eight destinations, Panama City, San Jose (CR), Roatan, San Pedro Sulas, Guatemala City, Managua, Cancun, and Madrid, along with one-stop Barcelona and Tel Aviv. Each flight is daily except Madrid, which is twice daily (and a third codeshare with a AA) and TLV, which is thrice weekly. Air France has a 737 in Miami operating MIA-PAP-FDF-PTP and back.