A wire antenna that doesn't have a specific geometry may be called a random wire antenna. Such an antenna will correspondingly have a random performance with a random radiation pattern. In these cases, typically we care more about getting any signal out at all, rather than doing it efficiently or in a specific direction. A random wire antenna might be cut to length for a specific frequency, or it may be a more or less random length with an antenna tuner between it and the radio to match the impedance. (Even a random length wire may need to be cut within a range of lengths for some tuners to be able to tune it.)
A dipole's most important characteristic is that it is close to a half wave long. It's geometry isn't critical beyond that, but variations in geometry will change characteristic impedance and radiation pattern (and possibly polarization) at least. Typically a dipole is depicted as two quarter wave segments that are colinear. Practically, we may want to actually have them in an inverted V pattern for best impedance, but other patterns work as well, and may or may not affect radiation pattern significantly. A dipole has some directionality, but typically the concern with a dipole is polarization (for vhf) and elevation angle of the radiation pattern (for vhf and hf). For HF dipoles, height above ground may affect elevation angle of the radiation pattern more than the antenna's internal geometry.
A dish is not an antenna, but may be part of an antenna system. A dish is a reflector, and typically has an antenna near the focus of the dish with a radiation pattern that covers the dish; the dish shape is critical to focusing RF energy on the antenna, and the antenna's shape is important for optimizing the aiming of the energy at the dish. The dish is typically parabolic but may be spherical or hyperbolic, and the curve may be in 2 or 3 dimensions. The ideal antenna for the dish can be any number of models with a hemispherical or one sided conical radiation pattern. (Although, a 2d parabolic dish could use a dipole exactly at the focus, parallel to the axis of curvature.)
Other antennas, for example, yagi, moxon, patch, etc., are directional antennas, and their geometry is critical to forming their directionality. The directionality also is coupled with making the antenna high gain.