My first, and only, direct experience with homeopathy was a bottle of teething tablets I bought on the advice of a friend. That was a few years back. I didn't look closely at them at the time since they came recommended. The did kind of seem to work because whenever the baby was crying and we put one in his mouth he immediately stopped crying. Sometimes that was enough to coax him back to sleep, but often the effect wore off before that. In those cases typically multiple tablets didn't help.
So what does that mean scientifically? Bupkis. I have nothing to compare the tablets to and I have no idea whether they worked any better than a simple sugar pill would have. That's not science, that's an anecdote. Correlation is not causation, etc., etc.
Instead, here's a great article about homeopathy covering its history, theory and state of scientific research. The long and short of it is that to date no convincing evidence has been found to show that homeopathy is anything more than well shaken water. When that occurs it's only logical to conclude that it is in fact not a valid medicine. I mean, at some point we have to move on to other research and leave the losers behind.
I ended up throwing away the teething tablets before the next baby came along. I can't recall whether we ran out or I just chucked them on principle. Either way, I shan't be purchasing any more.