I would create separate estimates for volume and choice of debt instrument. There are tools to estimate these simultaneously but I do not see a compelling advantage here.
I assume the volume is conditional on the choice of debt issuance so you might start by predicting choice of debt issuance and use this as an input to the volume model. The volume model would have a dummy variable associated with the type of debt issuance (you need k-1 dummy variables where 'k' is the number of debt issuance types). You could choose to allocate the probabilities across each dummy variables, or predict the volume conditional on the type of debt issuance.
For volume, there is a broad variety of regression models and ways to proceed. I will illustrate a conventional approach:
1) Develop hypotheses on which variables/factors drive your dependent variable.
2) Consider normalizing your variables (z-scores).
3) Identify interactions amongst variables - particularly economic variables.
4) Add dummy variables to represent covenants.
5) Perform variable selection if necessary.
6) Train a model and validate on a hold-out population to guard against over-fitting.
Note there are quite a few details in the process : missing imputation, outlier treatment, variable selection, seasonal & ARCH effects in traded volume, and functional form specification that are beyond the scope of this post.
It may be preferable to model volume for each type or segment debt issuance -- for example, I imagine US treasuries have the most volume traded when high-yield bond markets are frozen (flight to safety). Also, the model dynamics may vary based on the economic state variables. If that is the case, consider a regime switching model such as a dynamic linear model (for continuous states) or hidden markov model (for discrete states).
To estimate the probability of a particular debt instrument a multinomial model would work - so would k-means clustering, CART classification, SVM classification, or a variety of other tools. You could make a more informed decision among these if you had some theory relating attributes to your dependent variables. In the absence of such a theory, an empirical examination informed by your hypotheses would likely reveal some insights.
You will also want to consider the cost of mis-classification as you evaluate the confusion matrix from your classification model and tune this parameter in your classification model.