Yesterday's daf (Shabbos 43) has the din of מבטל כלי מהיכנו, which means that you take a כלי and put some מוקצה into it rendering the כלי unfit for use the rest of Shabbos. Rashi on the mishna (42b) says that the prohibition is based on בונה. Since you are causing the כלי to be fixed in place (you aren't allowed to move מוקצה) it looks like בונה. Rashi later on in Shabbos states that it is based on סותר. You are taking a כלי which is fit for use on shabbos and making it unfit for shabbos.
Based on this, we have the following problem. Most people take out the garbage before Shabbos. If the garbage can is empty on Shabbos, then when you put garbage in on Shabbos it should be a problem of מבטל כלי מהיכנו. You are taking the garbage can which is a כלי and putting מוקצה into it. Therefore, some people are machmir to make sure to put in a piece of garbage before Shabbos.
The minhag is to be מיקל, what is the basis? Many say that since the garbage can is made to put garbage into it and that is it's whole purpose it is not called מבטל כלי מהיכנו. This makes a lot of sense if the prohibition is based on סותר. Since putting garbage into the can is it's purpose in life you can't call it סותר, you aren't destroying the כלי you are using it. However, if the prohibition is based on בונה this shouldn't matter, you are making the garbage can unmovable.
The bottom line is that most poskim are מיקל.