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?

I'm a fan of exploratory data analysis, but you should be cautious of it and the intended variable selection.  If you make any decisions about which variables to use or how to model the data based on what you see in the data, the resulting model will be biased and any inferences (assuming that's of interest) would be invalid.  One possibility is to randomly partition the data into two subsets, explore one, and pursue your chosen model on the other.  If you're not interested in inference, you could use a LASSO model to get variable selection.
The shapes you see in these plots are common for count data, which are typically highly skewed.  Furthermore, it isn't at all clear that there are any actual 'outliers'.  Individual points that lie beyond the bulk of the data are normal for count data like these.  These plots do look like they show potential relationships.
To help you view them, I might suggest trying several things:

*

*Experiment with some transformations.  I wouldn't be surprised if log transformations are appropriate for both X and Y.  (Remember, though, that a Poisson GLM uses a transformation of the conditional mean, not a transformation of the data, so this won't be quite the same.)

*The resulting plots will still have stripes.  If the skews and variances have stabilized, you could try adding a small amount of jitter to nudge the data apart.

*Likewise, making the data semi-transparent may similarly help.

*Lastly, it may be beneficial to plot a LOWESS line over the scatterplot to help see the relationship.

