They really convey different meanings, so it depends on the purpose. It is useful to have the standard deviation so the reader can get an idea of how spread out the data are (and depending on your formula the SD is an estimate of the population SD ... and the sample SD formula is close enough unless your sample is small). If you are emphasizing the precision of your mean estimate, then the SE or a confidence interval would be more useful.  You can present both. This issue is also important when you are deciding what plots to do (and again you can include both in many plots).

The SD can be used as a description of the observed data sample and as an estimate for what to expect for the statistical population.

The SD refers to the "spread of the data", whereas the SEM refers to the "spread of sample means". The SD is interpretable (in the usualy way) when the distribution of the data is approximately normal: if this is the case, then we know that there is a 68% probability that a randomly selected/measured value will fall in the range mean±SD, and that there is a 95% probability that it will fall within mean±2SD, 99% within 3SD etc. If the distribution of the data is not normal, one can calculate the SD, but its meaning will be different and it is not so clear how to interpret a particular value.

The same applies in principle to the SEM. If the sample means are approximately normal distributed, then we know that there is a 68% probability that a random sample mean will fall within mean±SEM, a 95% probabiliyt to fall within mean±2SEM and a 99% probability to fall within mean±3SEM etc. This can be used to construct hypothesis tests and confidence intervals around the sample means (this procedure has a defined probability of generating an interval that includes the mean of the statistical population). In contrast to the SD which relies on the distribution of the data, the SEM relies on the distribution of the sample means. The central limit theorem says that with increasing sample size the distribution of the sample means will approximate the normal distribution. Thus, for large samples the SEM will be interpretable (as stated above), no matter what the distribution of the data is. For small samples one has to take extra care, not only w.r.t. to the distribution but also w.r.t. the construction of tests and confidence intervals (that will then be based on the quantiles of the t-distribution instead as of the normal distribution).

What to show/report?

If you want to show how the data varies, you should present the SD - if your data is approximately normal distributed. If this is not the case, reporting the IQR is preferrable.

If you want to show how sensitively the likelihood of your data depends on the estimated mean, then you should report the SEM (this makes sense only when the distribution of the sample means is approximately normal; otherwise the meaning of the SEM is hard to interpret, and the common interpretation is misleading).

Forgive my ignorance.

But would you say that "spread of sample means" is the standard deviation of the standard deviation?

@Terun:

no, that's not correct.

The sample SDs will have their own distribution, which is *not* the distribution of the sample means. These are entirely different things. One could estimate the standard deviation of the distribution of sample SDs, what would be the "standard error of the SD" (not the "standard error of the mean"). Note that the distribution of the SDs is unlikely to be approximately normal, so interpreting such a value is not straight-forward.

Also note that if we make inference about the mean based on an estimation of the SD* and the assumption of normal distributed values, the uncertainty in the SD is essentially propagated by using the t-distribution instead of the normal distribution.

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*actually the variance is used, what is the square of the SD.

The standard deviation is a truly awful measure of the spread of data. It measures spread around the mean. However, spread is often asymmetrical around the mean. Family size, for example, or depression scores, length of hospital stay – I am surrounded by data that are simply not symmetrical, let alone 'normal'. 

Most people don't read the standard deviation or, if they do, worse still, come to erroneous conclusions about the data based on it. For example, I'm working on a dataset right now with an outcome variable that has a mean of 4·2 and an SD of 2. What does this mean?  

People may tell you that most of the values are expected to lie between 4·2±2SD, that is  between 0·2 and 8·2. However, this particular scale only goes from 1 to 6. Values below 1 or above 6 are undefined. 

Knowing that, people might mutter about the data being skewed (which they are), but it's no substitute for a proper description of the data. 

Quartiles, quantiles of interest, graphical displays are all useful in describing your data. The standard deviation rarely is. 

For RCT usually effect size reported (odds ratio, risk ratio - for binary outcome, shift in means - for qualitative primary variables with 95%CI). It is first interest for clinicans.

RCT's often are multicentred.

To provide meta-analysis - is needed (groups) sample size, estimation primary variable in groups.

With respect to persons who execute meta-analysis mean and SD is more preferable (for qualitative variable). For binary outcome - proportions in groups. Count variable ( stay at hospital, for example) - median, range, IQR

Thus*, the size of the effect = with SE or 95%CI

Groups descriptions =with SD

* It's only my expirience with clinicans and analysts.

There is some guide: European Medicines Agency, Topic E 9, Statistical Principles for Clinical Trials, may be useful for Your RCT.

I typically report both. The SD is presented with the summary statistics. It is a summary of the spread in the observations; for comparison of groups, this is important, especially when the sample sizes are different. Also when the sample sizes are large, SE may be dominated by the sample size and there will appear to little variability. On the other hand, the SE is essential for inference, so I include the SE of the estimate used for inference, such as Least Squares Mean for ANOVA

@David

You mention SE is essential for inference. How does SE tell you this?

I meant that the SE is essential for inference because the statistical inference (testing, confidence intervals, etc.) depends on SD only through SE.