What is the purpose of sifting dry ingredients?

Fix Compacted Flour. Flour will compact over time (and during shipment). You could sift the flour to fluff it back up. Or, you could just stir it before measuring and be sure to spoon the flour into your measuring cup in order to get a correct volume measurement.

Remove Unwanted Material. Yes, sifting would also remove larger pieces or bits of chaff. It would also remove insects. However, neither of these are problems with modern store-bought flour. If you grind your own flour (my sister-in-law does this), then you may still want to sift it, though.

Mix Ingredients Together. Sifting can also be used to mix other dry ingredients into flour. I bake quite a bit, though, and I've never had a problem with just using a spoon or whisk to mix dry ingredients together.

In summary: don't bother. Just use good measuring techniques and stir your ingredients together well. No one likes a lump of baking powder lurking in a muffin!

Sifting aerates the flour. This alters the texture of the finished good, resulting in a lighter, airier texture. This still has relevance today, and should be done when the recipe calls for it, but you can experiment freely on your own. Generally there is no need to sift when making bread, biscuits or scones. Delicate sponge or chiffon cakes using pastry or cake flour must be sifted given the proclivity for clumping these flours have.

You should be aware that pulsing your dry ingredients in a food processor is a bit more effective at aerating, as well as mixing the ingredients. It also requires much less effort.

I use a whisk in a bowl to mix dry ingredients before adding them to wet ingredients when there's a huge difference in the volume between some of them. For example, adding a few teaspoons total of soda, salt, spices, etc., to two cups of flour. I think this does a good job of distributing the ingredients and I don't care much for using an actual sifter.

Also, for recipes that call for sifting flour, etc., before measuring by volume, you will get a vastly different amount of that ingredient compared to measuring without sifting it. You can cheat this by using weight instead of volume.

According to Cooks Illustrated:

Sifting flour or cocoa powder is a chore, but sometimes it is important. When making a delicate cake like a sponge cake or genoise that requires flour to be folded into beaten eggs and sugar, sifted flour can be added quickly and distributed evenly (because sifting aerates the flour), thereby reducing the risk of deflating the batter. Recipes with cocoa powder, such as chocolate cake, also often call for sifting the cocoa powder. In this case, sifting breaks up small clumps of cocoa that form as the powder sits in the package. Sifted cocoa can be evenly distributed throughout a cake batter; with unsifted cocoa this isn’t always the case.

The main purpose of sifting of dry materials through a sieve is to remove all foreign particles or to break up lumps.