Icicles can be a good indication of one of two things, or a combination of both.
1. Lack of insulation. Due to insufficient insulation, heat loss will warm the attic and will allow residual melting from underneath.
2. Lack of ventilation. Due to insufficient ventilation or a balanced ventilation system, any heat loss or radiant heat from the sun stays in the attic and allows the snow to melt from underneath. With inadequate ventilation, many times moist air is trapped in the attic causing condensation on nails and plywood or roof decking. In many cases, this causes damp or wet insulation and can create mold or mildew in the attic.
In both cases, as the snow melts from underneath the water (melted snow) runs down the roof until it gets to the unheated (usually the eave) portion of the roof and will freeze. As this continues, this is what fills gutters or makes ice dams on the eaves.
Once the ice dams, the water will continue to run making a larger dam, or in many cases the water will pond behind the ice dam over the heated part of the house and cause leaks. Roof shingles were meant and designed to shed water not pond it.
In some cases, even with ice and water barriers underneath you can still have water leakage. For example, through a vent pipe boot or when the water in some cases backs up further than the protective barrier.
If you see icicles, please consult your professional roofing contractor and/or insulation professional.