Verticals covers all the common types with a coil calculator built in for the antennas that require them.
The quagi may be the best of two worlds. It has the yagis gain and the quads direct feed and signal capture. I could not find any published formula for them but only sample antennas. From these samples, and computer modelling, I developed the program. I have only built a 440mhz version which works very well but would like to hear about your results.
The discone is most seen as a receive antenna but is under-rated. It can cover multiple bands and has more gain than verticals. This also is true of transmitting. It does have a variance across its range that can't be easily predicted so if the match is not what you want, just increase or decrease the starting range to shift the "nulls". HF versions have been built that cover many bands.

 
The Skyhopper was named after the first one was compared to a grasshopper . The antenna is basically a quad opened up on the bottom. This up and over, over and down design had been used in HF wire antennas. The antenna acts like a stacked pair on the vertical and a standard yagi on horizontal. The prototype for 2 meters made its first contact of over 200 miles while lying upside down on a picnic table. This lead to the discovery that it can be used very low to the gound by inverting it. This method can also be used when there is match problems due to nearby objects. A 440 version with suction cups allowed an apartment dweller to have a beam antenna stuck to his ceiling. The antenna is very forgiving of dimensions. The prototype had screws in the end of the elements for adjusting but at 2 meters, 1/4 inch changes did not affect performance. I highly recommend this for new builders. The only problem is that it fires off to the corner opposite the feed. Over the years I have seen the design renamed and modified by others, and posted on other web sites.This is the original.

Parabolic antennas are the most directional of all antennas and have the most gain. Because of their size, they are usually used for UHF and up. They are much easier to built than you think. Any long bar will bend to a parabolic arc when stressed on the end. If you mount 1 X 1 inch wood strips solidly to a center hub, a wire can be strung through loops on the outer ends. When the wire is drawn tight, the ends will curl up forming the dish.

FOR THE EXPERIMENTER
  
Homebrew capacitors, gamma match  and homebrew feedline.
If you don't have the means for measuring reactance, resistance and capactance on your antenna, the gamma program will give you a starting point for for designing the gamma match.

Single line next to the tower, twin ladder line, 4 wire line and even HOMEBREW HARD LINE can be designed with the Feedline program.



Back in the days when CB was at its height, with CBers in the millions, the antenna companies strived to build the best. Two interesting designs emerged. Avanti Antennas produced a little quad called the PDL-II. This quad , fed by 2 coax lines, was able to work both vertical and horizontal. Its "co-inductive" secret was that it used a folded dipole to feed the quad. This gave the antenna more gain than a simple quad. I modelled the program to find out how it worked. The loops of folded antenna have been replaced by V shaped elements without any changes and easier matching. This antenna is worth experimenting with especially with its reduced size. Co-inductive and PDL-II are trademarks and this antenna is most certainly covered by patents.You are allowed to build a not for monetary gain, experimental version for yourself.


The other antenna was made by Antenna Specialist who are still in business and may still produce the Super Scanner. This antenna beams into three quadrants without the use of a rotator. By using relays, a single feed line could be switched to 1 of 3 elements or to all for an omni position.The coax jumpers made the other 2 unactive elements appear to be longer and act as reflectors. Later versions grounded the inactive elements through a tuned capacitance/inductance circuit. I found that the jumpers work better at times when made of 75 ohm coax. This antenna is recommended for advanced antenna builders only as it is VERY sensitive to element length and spacing changes. Some adjusting will probably be required. This antenna and the name SuperScanner have patents and trademark restrictions.
No patent, trademark or any other type of infringement is intended against the Antenna Specialist Company!


NEW
As soon as I saw QUAD in Quadrifilar Helicoildal Antenna, I had to investigate it. Used for weather satellite reception, this omni directional antenna has everything you want. It is small in size, has a very low angle of radiation and up to twice the normal verticals gain.

The program has 6 variations of the design,  with different formulae for each. This is the most interesting design I have ever researched. After 200 hours, the program is still in the research stages but I have discovered many points about its operation and decided to post what I have.

There is no construction or matching details in the program yet, so check out qha sites on the net for this info. I built a 2 meter desk top version out of foam disks and 20 awg wire. Wrapping the coax around the 3/8 fiberglass pole about ten times was good enough to reduce the match to 1.5 over a mhz. It measures at 5.1 db gain and I can work all the local repeaters and about 15 miles simplex with the antenna sitting on the BASEMENT floor. It also appears to work like a discone on receive.
Receive is good outside the band both above and below the 2 meters.
It easily hears from 137 to 174 mhz. The use of small wire instead of tubing did not affect the bandwidth but it did shift the frequency down about 3/4 of a mhz. To correct this, I found that shortening the large loop shifted the frequency much more than changing the small loop. THIS IS VERSION 3 ...... All except the RCA version are designed to be made out of 3/8 inch tubing which is easier to work with.
The dimensions are NOT designed for use with elbows or bent radius corners. Cut the tubing 1 diameter longer than the required length , flatten the ends in a vise and drill bolt holes at the correct dimensions. After assembly you may still solder the joints. All are designed to be used 5 meters or more above ground. Patterns and phasing are lost if you place them on the ground. Using an A/D converter hooked up between the receiver and a computer plotted out the receive pattern of the 1/2 turn .3 ratio Wxsat Qha mounted 20 feet up on the rooftop. The pattern was almost identical to the modelling results shown on the patterns page. The signal is solid from horizon to horizon , being out only seconds from predicted passes.
 NEW Matching calculator for low impedance QHA ...HERE