Given a choice, I will always choose to shoot with a tripod. However, there are many public places, like cathedrals and public gardens (like Longwood Gardens, main conservatory shown here), where tripods are simply not allowed. In those cases I almost always bring a monopod to steady the camera. A monopod is essentially a one-legged tripod (your two legs provide the other legs of a tripod) and while not as steady as a tripod, they can help you shoot at much slower shutter speeds than you would normally shoot handheld (even with vibration reduction).
A good monopod (I have a Bogen/Manfrotto) will cost you under $100 and you can add a small ballhead with a quick release for about another $100. Every time I use my monopod I bless the day I started using them. Not only do they steady the camera nicely, but they support the weight of longer lenses (taking the weight off my shoulders) and let me stand "at ease" without the weight of the camera strap pulling on my neck--this adds hours to my shooting stamina.
When it comes to choosing your monopod, put some weight (camera weight and your body weight) on it to be sure it won't slip in use. I have monopods that are 20 years old, so try not to cheap out; years from now you'll be glad you spent the extra few dollars for a top-quality monopod.
3 days ago