What is the SINGLE most appropriate investigation for a 55-year-old man with frequent urination and a deep painless ulcer?

The most appropriate investigation for a 55-year-old man presenting with frequent urination and a deep painless ulcer is blood sugar testing. This choice is pivotal due to the clinical implications of the symptoms described.

Frequent urination can be a classic symptom of hyperglycemia, particularly in the context of undiagnosed or poorly managed diabetes mellitus. Additionally, a deep painless ulcer, especially in an older individual, raises the suspicion of diabetic foot ulcers, which are complications that can arise when diabetes affects vascular and nerve function. Evaluating blood sugar levels can help determine if there is an underlying metabolic disorder such as diabetes, which could explain both symptoms.

In contrast, the other investigations do not align as well with the initial presentation. Arteriography and venography would typically be used to assess vascular conditions but do not directly address the symptoms presented. Conducting a biopsy for malignant melanoma could be indicated in the case of a suspicious skin lesion, yet without more context suggesting malignancy, it wouldn’t be the first investigation to pursue when the symptoms point more towards a metabolic issue like diabetes. Therefore, measuring blood sugar levels is the most logical and direct approach to assess the underlying cause of the patient's symptoms.