Which was the last significant event in human evolution? What's next?
Background

I'd like to start by saying that phenotypic (loosely speaking phenotypic mean morphological) traits evolve because their variance in the population is correlated with some genetic variance. If most of the phenotypic variance is genetically coded, you'd expect that the trait will change through if different variant of the trait influence fitness (which is a function of both survival and reproductive success). Stated differently and more accurately the change in mean value of a given trait $R$ in the population is given by:

$$R=S\frac{V_{Ga}}{V_{P}} = S\frac{V_{Ga}}{V_{G}+V_{E}} =  S\frac{V_{Ga}}{V_{Ga} + V_{Gd}+V_{E}} $$

, where $S$ is the selection differential, $V_{G} = V_{Ga}+ V_{Gd}$ is the genetic variance where its two components are the additive and dominance genetic variance respectively. $V_P  = V_G + V_E$ is the phenotypic variance which two components are the genetic and environmental variances. The ratio $\frac{V_{Ga}}{V_{P}}$ has a name; it is called the heritability in the narrow sense symbolized by $h_N$ Note: I present here a simplified version of the reality! 

So, if the trait you consider has genetic variance in the population and that this trait influence the fitness with a given selection differential you can provide estimates of how the trait will change through time in the human population. Now if the variance does not already exist in the population but you have to wait for mutations to occur, then you cannot really provide an estimate of when will this occur.

Discussion on the future evolution of nail size

Also, when you think about some traits such as our nails. You may consider that they are of no use but do you really think that those people that have no nail would really have a better fitness. At first I would tend to think that I would rather chose a partner that have nails, it is much more sexy! If many people are like me and appreciate nails, nailless people will have difficulties to reproduce or in other words, they will have a lower fitness that nailed people.

I don't think we have data on what loci (positions on the chromosomes) influence the size of the nails and about how much of the variance in nail size is explained by the genetic additive variance. And how do the size of our nails influence fitness. So we don't know about how this trait will evolve through time.

Future evolution of other traits

There are several traits that we can/could more or less predict their evolution. Those are the traits for which we know the ratio of the genetic additive variance over the phenotypic variance and how it influences fitness. I can think of genes that are linked with resistance to new diseases such as HIV for example. If malaria comes to a new area you might expect the allele (=variant of a gene) allowing a better resistance to malaria will increase in frequency (note that this allele is advantageous in heterozyotes (protects from malaria) but it is highly disadvantegous in homozygote because it causes sickle-cel disease). We could estimate the change in frequency of alleles linked with obesity in some countries also. Probably several studies exist on the subject but I don't know them. One might improve this answer by giving some information about how HIV resistant variants is likely to change in frequency through time.

What were the last evolutionary changes in human history?

You can think of the impressive frequency of the Tay-Sachs disease in french canadian due to a bottleneck effect. Or you may think of the evolution of height, about the evolution of the shape of the skull or the evolution of feet. You can think of the reduction of our gut. etc… Your question is not well enough define so that we know if you consider the raise in frequency of Tay-Sachs disease is sufficient for you to be called a significant evolution.

Hope that helps!