What blockbuster class of drugs is taken by about 20% of all people over 45?  This is obviously a huge chunk of the population and so the first question should start with their safety and side effect profiles.
The answer is revealed all too quickly in their advertisements. For example, have you ever looked at an add for Lipitor, Pfizer's blockbuster statin for people with high cholesterol? The most common side effects - and keep in mind this is published by Pfizer itself - are headache, diarrhea, gas, rash, constipation, upset stomach, stomach pain, muscle aches and joint pain. The ad goes on to say that "only" 3 people of 100 will stop taking Lipitor because of side effects.
Now I do have to say that all 3 Plaque Regessers as I call them use statins to one degree or another:
a) Dr. Gould: I have the impression he uses a statin with most of his patients.
b) Dr. Esselstyn: He will use a statin if a low fat diet does not get cholesterol below 150, but this is in a small minority of his patients. (These men may be hypothyroid or overweight?)
c) Dr. Davis: Occasionally uses a statin, but finds them generally unnecessary. (He uses high dose niacin.)
That said, I think statin use is risky business and I am doing all I I can to lower my LDL/cholesterol naturally without have to resort to a statin. For those interested, I have some pages that may be of interest: How to Clear Your Arteries and How to Lower Your LDL Levels Naturally. My LDL is the 70's and my cholesterol has long been between 125 and 135 without any medications or even supplements. This page covers how I eat: The High Protein Low Fat Diet.
Here are just some of The Dangers of Statins uncovered by the research:
1. Lowered CoQ10 Levels. One of our bodies master antioxidants is CoQ10. Our mitochondria, the energy burning furnaces of our cells, spit out free radicals and CoQ10 is part of our natural protective mechanisms. Statins go in and lower the activity of the enzyme immediately above our body manufacturing of Acetyl-CoA. In fact, statins are called HMG-CoA Reductase Inhibitors due to the enzyme that they act upon in this process.
2. Lowered Testosterone. Furthermore, one recent large study of almost 3,500 men with an average age of 51 found that men on statins were twice as likely to have low testosterone.  The lead researcher speculated that this may be due to a decrease in pituitary output. I should mention, though, that the research shows only a minimal change in testosterone, something I document in my page on Testosterone and Statins. However, statins can produce a low testosterone type of effect in some cases, which I discuss in that link.
3. Increased Risk of Erectile Dysfunction. Just as bad is that statins have been found to be associated with erectile dysfunction in some studies.  Zocor is likely one of the worst offenders.  One study even comments that the erectile dysfunction reported seems to exceed the simple explanation of lower testosterone levels.  Lowered libido is another commonly reported side effect. (In fairness, some study work shows improved erectile function in high cholesterol men.)
4. Increase in Oxidized LDL. Statins are designed to lower LDL and, in general, lowering LDL will help to arrest or reverse the buildup of plaque all things being equal. However, with statins not all things are equal: they increase oxidized LDL probably due to the decrease in CoQ10 mentioned in #1.  Oxidizing LDL is essentially turbocharging this lipid to do damage to your arterial walls and could easily increase arterial plaque. Yes, you have less LDL, but that LDL is more likely to cause damage thanks to the lack of CoQ10. Theoretically, you can just take some CoQ10 and this may help, but it begs the question as to whether this is the right approach for most men.
5. Increased Arachidonic Acid. Anyone who has read even a little about health knows that cooling down inflammation is one of the major goals of any man living a modern lifestyle. Our Western way of living is almost designed to increase inflammation and inflammation has been linked to heart disease, cancer, arthritis, Alzheimers - the list goes on and on. One of the ways to increase inflammation in the body is to increase arachidonic acid and so consider what these authors stated:
"It has been shown that statins enhance arachidonic acid biosynthesis from linoleic acid in vitro, and there is also evidence that statin-treated patients have high plasma and cellular arachidonic acid levels." 
The consequences of this are difficult to tease out, and I should point out that some statins are considered anti-inflammatory actually.
6. Myopathy and Muscle Damage. For reasons that remain elusive, statins can cause muscle pain, weakness and even damage in around 2% of patients. (Some say more.) The side effects can be very serious indeed: a recent study found that 25 out of 44 of statin users with myopathy (persistent muscle pain) has actual damage to their muscle fibers of 2% or more. The damage remained at least for a time even after the statins were discontinued. Scientists are still unsure what causes this damage but it may be that statins whack CoQ10 levels, an essential molecule and key antioxidant.
7. Kidney Damage When Used with Certain Antibiotics / Antifungals. There is evidence that statins, when used with certain antibiotics (clarithromycin/Biaxin for example) and certain antifungals, can cause actual kidney damage.
8. Increased Risk of Diabetes. A couple of studies had indicated that statins increase diabetes risk. However, these studies were criticized for a variety of reasons. However, a recent study tried to overcome all these concerns with multivariate analysis and controlling for many key variables. The results again showed an almost 50% increased risk of developing diabetes. Furthermore, the authors clearly identified a root cause of the issue as well:
""Furthermore, decreases in insulin sensitivity and insulin secretion were dose-dependent for simvastatin and atorvastatin. And, after adjustment for all the confounders mentioned above, high-dose simvastatin was associated with a 44% increased risk of developing diabetes, while for low-dose simvastatin the increased risk was 28% and for high-dose atorvastatin the increased risk was 37%. Overall, 29% of participants were taking simvastatin, while 53% were taking atorvastatin." 
9. Sleep Disturbances. Zocor has been found to lead to decreased quality of sleep according to one study.  The probably lies in the fact that Zocor (simvastatin) is lipid soluble, which allows it to easily pentrate into the brain. The brain is very sensitive and incredibly finely tuned. The same study showed that this medication actually altered sleep patterns, obviously not a good idea.
10. Weight Gain. A recent study found that "patients taking statins experienced a modest 12% increased risk of developing type 2 diabetes over a four-year period, and gained around half a pound (240 grams) in weight on average."  Admittedly, this isn't a large gain, but is a sign that statins shift things in the wrong metabolic direction.
11. Increased Risk of Parkinson's Disease. A large, recent study found very ugly results - a very significantly increased risk of Parkinson's from statin use: 
"Of the subjects, 21,559 were identified as having PD on the basis of criteria of having a primary or secondary diagnosis, using anti-Parkinson's medication, or having deep-brain stimulation surgery." "In the cross-sectional analysis, the use of cholesterol-lowering drugs, including statins or nonstatins, was associated with a significantly higher prevalence of Parkinson's disease (odds ratio [OR], 1.61 - 1.67; P < .0001) after adjustment for age, sex, and other comorbidities, such as hyperlipidemia, diabetes, hypertension, and coronary artery disease."
SUMMARY: As cautious as I am about statins, do not quit without consulting with your doctor. Also, I would like to mention that targeting LDL alone is not a safe strategy. In general, you need to also target HDL and triglycerides. I cover this in my page on HDL, LDL and Triglyceride Levels to Reverse Atherosclerosis. This helps insure that your not low LDL and prediabetic at the same time, which can increase your LDL Particle Count (LDL-P). HDL and triglycerides are powerfully affected by your glucose/insulin control or lack thereof. Better yet, get a VAP or NMR lipid blood test and pull your LDL-P. LDL-P is actually the number most predictive of arterial plaque and heart disease. See my page LDL-P and Arterial Plaque Reversal for more details.
A NATURAL WAY TO DROP CHOLESTEROL: I am partial to whole food, low glycemic lower fat diets. One study showed that these maintain current testosterone levels and will halve your estradiol levels. See my page on The Testosterone-to-Estrogen Ratio for the research. In addition, a well done low glycemic low fat diet will lower your LDL dramatically and give you very good triglyceride numbers as well. Inflammation is dramatically improved, instead of worsened and oxidized LDL will be dramatically improved. Ntiric oxide levels will be increased and blood flow increased. Basically, a well done low fat diet does everything that statins do in my opinion but without all the side effects. The key is that it must be low glycemic. See my page on How to Reverse Diabetes with a Low Fat Diet for more information.
One additional comment is that some researchers have positied that lower fat diets could increase the risk of Parkinson's. First of all, my dietary fat levels are in the 15-20% range, which is not super low fat. More importantly, many of the longest-lived and healthiest cultures on the planet have very low cholesterol levels (by American standards) and have no Parkinson's, Alzheimer's, cancer, heart disease and so on. In fact, EVERY primal culture that Loren Cordain surveyed had cholesterol below 150, and all of these cultures are extremely brain healthy. So, clearly, it is not low cholesterol levels causing brain issues such as Parkinson's but rather low cholesterol working in conjunction with other lifestyle factors.
1) Corona, G. Journal of Sexual Medicine, April 2010
2) Consumer Reports on Health, Nov 2010, p. 8.
3) Drug Safety, Jul 1 2009 , 32:(7)591-597(7), "Statins and Erectile Dysfunction: Results of a Case/Non-Case Study using the French Pharmacovigilance System Database"
4) Urology, Apr 2008, 71(4):703-707, "Effects of Statin Treatment on Serum Sex Steroids Levels and Autonomic and Erectile Function"
5) Family Practice, 2001, 19:95-98, "Do lipid-lowering drugs cause erectile dysfunction? A systematic review"
6) Family Practice 2002; 19(1):95-98, Do lipid lowering drugs cause erectile dysfunction?
8) Nutr Metab Cardiovasc Dis, 2001 Apr,11(2):88-94, "Statins enhance arachidonic acid synthesis in hypercholesterolemic patients"
12) http://www.medscape.com/viewarticle/870996 , "Statin Use Linked to Increased Parkinson's Risk", by Nancy A. Melville, October 26, 2016