Methods
There are several methods of livestock disposal to choose from, each has its own benefits and drawbacks. While there may be no correct answer for everyone, there are better approaches - especially as we factor in sustainability. Inspiration for this Q&A layout is drawn primarily from Amundson (2013). While this book concerns husbandry (see husbandry) of goats, I have tried to fill out her proposals to make them more applicable to a general audience (to include pet-disposal).
No part of this answer shall be construed as providing legal advice.
Incineration
This solution is often preferred where there has been large-scale contamination or infection by disease (see "Creutzfeldt–Jakob disease" in Rendering below). This method often results in an undesirable smell, and can be noxious to neighbors. Has low biosecurity risks, but true incineration must occur at high temperatures.
Within sustainable-frameworks, incineration could be combined with some type of energy-recovery from the waste-heat. Wikipedia offers some suggestions, but they seem more suited to industrial incinerators:
Incineration with energy recovery is one of several waste-to-energy
(WtE) technologies such as gasification, pyrolysis and anaerobic
digestion. While incineration and gasification technologies are
similar in principle, the energy product from incineration is
high-temperature heat whereas combustible gas is often the main energy
product from gasification. Incineration and gasification may also be
implemented without energy and materials recovery. {Wikipedia article, "Incinerate"}
In many jurisdictions, open-burning is prohibited.
Composting
A terrific treatment of meat-scrap composting has already been presented by @Jay-Bazuzi on this site for the question: "Why shouldn't meat be placed in my compost pile?" I will attempt to expand upon this answer to cover a larger scale (entire animal at-a-time).
Sawdust, straw, or other carbon should be increased to at least one foot (0.3 meter) underneath and surrounding the animal corpse. Straw is convenient because it might already be on-hand for animal bedding or plant insulation (to help a plant "winter").
Composting methods are often prescribed by law within the US, to include: primary and secondary composting (decomposition) bins, structural impermeability, setback distances from established water-sources, etc.
Livestock that perish from suspect neurological concerns are prohibited (at least in Illinois) from being composted and must be incinerated.
Burial
Another least-desirable method of animal disposal, second in my opinion to rendering, is burial. Burial is the unscientific dumping of an animal corpse beneath the ground. This method has a high potential for ground-water or well-water contamination. If buried too shallow, scavengers or pests will come to visit and potentially spread disease.
Rendering plants
Rendering is a process that converts waste animal tissue into stable,
value-added materials. Rendering can refer to any processing of animal
products into more useful materials, or more narrowly to the rendering
of whole animal fatty tissue into purified fats like lard or tallow.
Rendering can be carried out on an industrial, farm, or kitchen scale.
The majority of tissue processed comes from slaughterhouses, but also
includes restaurant grease and butcher shop trimmings, expired meat
from grocery stores, and the carcasses of euthanized and dead animals
from animal shelters, zoos and veterinarians. This material can
include the fatty tissue, bones, and offal, as well as entire
carcasses of animals condemned at slaughterhouses, and those that have
died on farms, in transit, etc. The most common animal sources are
beef, pork, sheep, and poultry.
While at first-glance, rendering might be seen as the embodiment of the reduce-reuse-recycle motto, biosecurity concerns have made rendering one of the least-sustainable disposal methods (in my opinion).
One of the most widely-covered, biosecurity events in recent times was the "Mad-Cow" disease outbreak. Properly called Bovine spongiform encephalopathy, this disease was spread to humans through contaminated beef which could not be rendered safe through cooking. When passed to humans, it is referred to as "variant Creutzfeldt–Jakob disease". The wiki article extract is below:
Creutzfeldt–Jakob disease or CJD is a degenerative neurological
disorder that is incurable and invariably fatal. CJD is at times
called a human form of mad cow disease (bovine spongiform
encephalopathy or BSE). However, given that BSE is believed to be the
cause of variant Creutzfeldt–Jakob (vCJD) disease in humans, the two
are often confused. {excerpt from Wikipedia} [emphasis mine]
References
- Amundson, Carol (2013). "How to Raise Goats: Everything you need to know." http://ow.ly/xBAAp
- Bazuzi, Jay. Untitled response. "Why shouldn't meat be placed in my compost pile?". Last accessed 7-Jun-2014.
- "Bovine spongiform encephalopathy," Wikipedia Article, http://en.wikipedia.org/wiki/Mad_cow_disease.
- "Creutzfeldt–Jakob disease," Wikipedia Article, http://en.wikipedia.org/wiki/Creutzfeldt%E2%80%93Jakob_disease.
- "Illinois Dead Animal Disposal Act," Title 8 of the Illinois State Code (CH. I, '90 SUBCHAPTER b), as retrieved from http://web.extension.illinois.edu/clmt/Workbook/WK_FILES/DEAD_A_W.DOC
- "Incinerate," Wikipedia Article, http://en.wikipedia.org/wiki/Incinerate.
- "Rendering," Wikipedia Article, http://en.wikipedia.org/wiki/Rendering_%28industrial%29.
- "Specified risk material," Wikipedia Article, http://en.wikipedia.org/wiki/Specified_risk_materials.