Actually, with the light weight metal gutters being sold these days and the wide spacing used for brackets (2 feet typically), it is not uncommon to see damage from ice buildup and glaciering of gutters, and downspouts ripping off walls because they used the recommended bracket spacing instead of 18" or so.
A number of ways to do it, pick the one that works for you - in each case, be sure water runoff is not ending up right by your foundation or otherwise causing damage like glaciering your walk or drive, and beaware with ladder work that steps willget wet while you work, so expect them to be icy.
1) pour hot water (full hot from the tap) in the gutter - and to get the downspout started if frozen solid at any point, fill top with hot water to full, then pour hot water on one side of outside of downspout to melt a small channel inside the pipe for the water from the top to start running down - usually will only take 1-2 gallons of hot water dribbled down the outside to thaw enough interior channel to start flow, then you can add hot water from top to melt out the rest.
2) wrap with heat tape, as prior suggestion said. Has to be low-temperature tape (not over about 120F) if on plastic downspout, to avoid melting it.
3) put hose on a faucet and run hot (but not full hot, to avoid melting hose) water and run water on outside of downspout to heat one edge, then run hose down into downspout with hot water to melt your way down, and from ladder put hot water into gutter starting at the downspout and working back toward high point while thawing enough to avoid blockage overflow, so it does not overflow on you and soak you and cause icing on the ladder. Then run water at high point of gutter till rest of ice frees up - takes FOREVER to totally melt with water, much faster to get it started enough to loosen up the ice in the gutter, then pull out lengths of ice by hand and throw them away. You will need heavy rubber gloves for this - I use this type with a fabric lining for warmth -
Pull hose back periodically to enlarge hole while in downspout so you do not get it stuck in the ice, as it will try to freeze in on you above the point where the nozzle is. If you are not into getting up on a ladder, you can buy a length to three (depending on gutter height) of 3/4 inch PVC pipe, three 90's, and a hose thread female fitting plus glue (or threaded PVC and fittings to allow disassembly for storage till next year), and make up an upside down J shape with the pipe that will put the water into the gutter, with the hose fitting on the bottom - only weighs about 5-10 pounds empty so easy to handle, and once hooked in gutter you can slide sideways without lifting entirely up. Disadvantage is you can't see how thawing is going, and of course if gutter overflows you have to move fast or get drenched with debris and icy water. BTW - 1/2" pipe willwork for about 10' (one length) - too flexible for greater height, get 3/4" for more height or if you anticipate using this to also clean out french drain or yard drain pipes or iced culvert under drive - all of which of course mean the J part at least has to be threaded for removal. Total cost under $50 including jet nozzle.
4) stick hose with jet nozzle like this (first two shown)
up into bottom of downspout, and shove up as it thaws from the bottom up - you need long rubber gloves for this or you will freeze your hands.
5) Use heat gun or blow dryer on a heat that is not too hot to touch (so you don't blister paint or melt plastic downspout) and heat downspout from bottom toward top till top blockage drains, then work around until ice inside breaks free and drops - very slow going if outside temp is low as pipe radiates heat away about as fast as you heat it. Do NOT stick in bottom of downspout to thaw - when ice or water breaks free and comes down you could be electrocuted.
6) Not great for plants below and promotes rust in steel and corrosion in aluminum downspouts, but you can dump about 2 soupcans full o of ice melt in top of downspout and on ice in gutter right next to it and let it melt its way down over the course of a day or two, then rinse salt away after thawed out. Slower than hot water, and not as good for the gutter or plant materials, but works. Ditto to ice in the gutter, though you will need several pounds for an entire gutter, and can kill grass in runout zone at that dosage.