I have been here for 6 months and have had zero issues (female traveling on my own). I started in Ecuador, then Peru, Bolivia, Argentina, a bit of Chile and now back up in Ecuador to hit Columbia in a week.
Out of all the places I have been.....Quito and Lima are the sketchiest. I was ALMOST robbed in Quito...dude loaded my bag under the bus as normal and then told me where to sit on teh bus...at the VERY back....I got a red flag and refused sitting near the front. He then tried to get a hold of my small backpack to put it underneath and I of course refused. Finally someone noticed and it turned out the dude did not even work for the bus company and his buddy was trying to steal my big pack underneath while he was trying to get my small pack on the bus. So beware.
I know a lot of ppl that have been robbed in Lima...and Quito (Mariscal area...aka Gringo Town).
I think the biggest fearfactor is on buses. I know of A LOT of ppl that have had stuff stolen on buses. Grip your bag tightly and guard it with your life. I sleep on buses a lot...can´t help it....so I just fall asleep with the bag all wrapped around my arms so it is hard to just grab and i would wake up if anyone grabbed at it. Never lay it beside you or on the floor or above your head, etc. Thieves are impressively good at what they do...they can reach behind the seat, under the seat, around the seat, etc without you even know it. Be careful. And of course pickpocketers are bad...I have only been pickpocketed once and they got a notebook probably thinking it was a wallet. I don´t know when it happened...they were good....
Cops are advised by law to not hassle gringos in Argentina...and Bolivia if I recall. Not sure about other countries. I have never had ANY issues with cops whatsoever. They usually do not bother you. Never allow a cop not in uniform to take anything from you....I have heard of this happening. Just refuse and go to a cop station...chances are they are fake.
I use my gut/intuition with taxis but having the hostel call for one is probably teh safest. I have refused some taxis cuz i get a bad vibe...or they are unmarked, etc.
I would also advise stitching on some secret pockets inside your pants and/or shirts. Stash money in a few places so that if you do get robbed, they likely won´t get it all. I also would advise taking your memory card out of your camera on buses so that if you do get your camera stolen...you won´t lose your pics. I always wear a money belt on buses but I don´t wear it otherwise. A friend was robbed at gunpoint on a bus in the afternoon in Ecuador...3 armed men robbed the entire bus. All they wanted was money so he was pretty lucky really....but still, scary and something you should be prepared for just in case. Hence the variety of pockets for stashing.
With all that being said, again, i have had no issues. Just keep aware of your surroundings...don´t walk down the street with headphones on or reading, etc....keep your head up and looking around. I tend to walk down the middle of streets later in the evenings too so that you are further out from little hideout places along the sidewalks. And, after having the near robbing experience in Ecuador, i pack around pepper spray just in case. I carry this in my hand if i feel uneasy about a place. Probably a bit overboard but it gives me some comfort knowing I have some way to protect myself.