"spikes" are only a real problem if you are soldering for high voltages. more heat and good flux will usually take care of spiking, spikes are a result of the solder cooling as you draw the iron away. for high voltage soldering, a blob technique is usually required where more than the normal amount of solder is used to form a ball that covers up any sharp ends of wires inside the joint. also there should be no bubbles or inclusions in the solder.
but for regular soldering, a small amount of spiking isn't a problem, it may not look "perfect", but as long as the solder flowed properly on the "work", it should be ok.