Appropriate visualizations for exploring count data I am preparing to do a regression model (probably Poisson) where the goal variable is count data (number of times a patient received a blood transfusion). I will have to do variable selection out of many available independent variables, and want to explore the data thoroughly before doing the actual regression.
I am accustomed to starting with histograms and scatterplots. But of course, the scatterplot shape is very different from a scatterplot with a continuous variable, consisting mostly of flat lines below the outliers.

For what it's worth, I did a univariate linear regression, and for both counts, it was highly significant, in the 10-7 and 10-6 region respectively. So if I were going off that information, I would expect to see at least some relationship visually, even though it won't be scattered around an actual straight line. But I don't.
How do I interpret this scatterplot? What better visualization are available for the relationship of the explanatory variable to count data?

In addition to the other great answers (use vertical jitter and transparency aka alpha to deal with overplotting!), you could consider 2d binning. It's a 2d analogue to a histogram.
In your case, only the $y$-axis seems to be discrete. I imagine you'd want to choose the vertical bins to be centered on the integers, and try several different horizontal bin widths to see which ones give a clearer visual representation of any trends in the data.
Since you seem to be using ggplot2, you could easily add a layer with geom_bin_2d() or geom_hex(). The former makes rectangular bins, while the latter's hexagonal bins are meant to avoid visual artifacts than might come from a regular grid. In your case, a regular grid is likely to be more appropriate since it would honestly show that only $y$ is discrete, but you could try it both ways.
It may also help to overlay the (transparent) points on top of the 2d bins, to distinguish bins with 1 observation from bins with just a few observations.
More generally, if both $x$ and $y$ are discrete, then also consider an approach like geom_count() which uses point size to represent the number of observations at each $(x,y)$ location.
