Convertible bonds usually have no voting rights until they are converted. Even after conversion, they may not be granted voting rights.
A convertible bond is a form of debt that features an embedded stock option allowing the convertible bondholder to convert his/her bond into a predetermined number of shares in the issuing company at some future date. There are many different kinds of convertible bonds, each with its own conversion features. Usually, convertible bonds are converted when the convertible bondholder chooses to do so - that is, the bondholder has the right, but not the obligation, to convert the bond on or before a set date. The terms of the bond's indenture detail the specific conversion clauses, such as how many and what kind of shares the bonds convert into.
In most cases, convertible bonds convert to shares of common stock, which usually have voting rights, but not always. Common stock is sometimes divided into different classes; when it's divided into two classes, it's known as dual class stock.
Typically, each class will have its own voting rights, and the voting rights of one class will be less effective than the votes of the other class (or classes) when votes are made.
For example, let's say you have convertible bonds of XYZ Computer Corp. that convert to 100 shares of Class A common stock, but it's the Class B common stock that controls 100% of the voting rights for the corporation (in this example). Class A common stock has no voting rights, which means that even after the convertible bonds are converted, you still don't have the right to vote on those issues that must be brought to shareholders for a vote according to the company's charter.
The list of different voting rights assigned to different classes of shares is endless. Another example of voting rights that give unequal voting power to different classes of shareholders are those that award a higher number of votes to shareholders that own one class of shares than to shareholders owning another class.
For more information on convertible bonds, see Convertible Bonds: An Introduction.
For more on voting rights, check out The Two Sides of Dual-Class Shares.