The expression "standard" of proof refers to the level of proof required, and is not to be confused with the expression "burden" of proof, which refers to which side has to prove an issue. The usage is, who has the burden, and to what standard?
There are at least two standards of proof in criminal law: proof beyond reasonable doubt, and proof on the balance of probabilities. Sometimes more standards are recognised: the standard of a reasonable doubt, as where a presumption applies, rebuttable by proof to the contrary (this is sometimes called, confusingly in the light of the usage I have set out above, the evidential "burden"); and a flexible approach to the balance of probabilities, where many standards or levels of proof might be recognised as appropriate depending on the seriousness of the issue. As to proof of a reasonable doubt, this is not recognised as a standard in New Zealand (R v Hansen, 19 September 2005 and 20 February 2007) but it is in the United Kingdom (R v Lambert 19 September 2005, 12 December 2008, and s 2 Drugs Act 2005[UK] which is yet to be brought into force by a Commencement Order). Other standards are sometimes referred to on appeals, where the appellant must satisfy the court that in the absence of an error at trial there was a "real chance" of a more favourable verdict. This "real chance" is less than on the balance of probabilities (Chamberlains v Lai 11 September 2006).
How do we know when a standard has been reached? At this point Montaigne again intrudes his mischievous little pen:
Courts are notoriously unwilling to assist juries by elaborating on the meaning of "beyond reasonable doubt". This is shameful. It is one of the law's little dishonesties that favour the prosecution. It is said that it is no use likening the standard to a given probability of proof, because people disagree on what the probabilities are. This is daft. The variation in what people think proof beyond reasonable doubt equates to in probabilities is astonishing (R v Wanhalla 25 August 2006). If people have not even a general idea of what beyond reasonable doubt means, how can they decide whether proof meets that standard?
Aversion to mathematical analogies is one of the law's shameful inadequacies. It reflects the stage in school when pupils, at age about 14, are divided into "arts" and "science" streams. Lawyers and judges are usually from an arts background, and have been brought up to think they cannot achieve an understanding of mathematics or science. At the same time, and ironically, lawyers and judges pride themselves on their logical reasoning. It will be interesting to see what moves are made in this area by the High Court of Australia now that it has a Chief Justice (Bob French) who has a science background. (Naturally Australian counsel are alert to the need for change: my thanks to Jeremy Gans for drawing my attention to Clarke v R  HCA Trans 336 (11 December 2009) in which special leave to appeal was refused by Crennan and Bell JJ.)
Areas of uncertainty exist. There is some doubt as to whether a standard of proof applies to a "risk", such as the risk of danger to the public if a person is released on bail (O v Crown Court at Harrow 31 July 2006) or on parole (Re McClean 19 July 2005). There is also doubt as to what, if any, standard of proof applies to proof of similar facts (R v Perrier and R v Holtz 14 October 2004; HML v R 26 April 2008). Sometimes the choice of standard appears inappropriate, such as the application of only the balance of probabilities to proof that evidence tendered to the court was not obtained by torture in another country (A v Secretary of State for the Home Department 9 December 2005), and there is still work to be done on what if any standard applies to the likelihood of unfair trial or of inhumane treatment in a foreign jurisdiction if the person is extradited (Gomes v Trinidad and Tobago 1 May 2009).
It seems intuitive that serious issues should require proof to a higher standard than issues that are less serious. This leads to difficulties where the law insists that a standard, usually the balance of probabilities, is fixed. The Supreme Court of Canada does not acknowledge that the standard is flexible, and it maintains that the degree of caution with which proof is assessed is the same whether the issue is serious or less so: FH v McDougall 19 October 2008. The opposite position is taken in the Strasbourg court: Saadi v Italy 11 December 2006, and this is probably the same as that taken in New Zealand: Z v Dental Complaints Assessment Committee 25 July 2008.
Another area where discussion of standard of proof has occurred is the fundamental requirement for evidence to be "relevant" before it can be admissible. Relevance usually means having a tendency to prove a matter of consequence to the determination of the proceedings. Variations on that formula occur, but the essence is "a tendency to prove". Serious discussions occur in the academic literature about what this means. What if an item of evidence has a small tendency to prove one thing, but a greater tendency to prove another? Is it relevant as evidence of the first thing? What if the item of evidence has only a very slight tendency to prove a matter in issue? In Bain v R 12 June 2009 the Supreme Court of New Zealand split on whether indistinct sounds were relevant on a recorded telephone call, the majority holding that to establish relevance it is necessary that the interpretation of the evidence advanced by its proponent be "reasonably" open or available. Further thought was given to relevance in Wi v R (see update to note for 4 July 2008) where the accused sought to adduce evidence that he had no previous convictions. This sort of problem calls for statistical analysis, but the New Zealand Supreme Court did not take that opportunity.